Ceratopsia, also spelled Ceratopia, is composed of those marginocephalians who are closer to Ceratops (or to the much better known Triceratops) than to Pachycephalosaurus. An important character all ceratopsians possess is the rostral bone, which gives the upper jaw the parrot-like beak characteristic of ceratopsians. Ceratopsians are almost all known from the Cretaceous of North America and Asia, with what appears to be a surprising amount of transcontinental travel.
    There are several groups of ceratopsians, including the basal psittacosaurids and the neoceratopsians (psittacosaurids cut out), consisting of the protoceratopsids, leptoceratopsids, and the ceratopsids. Also, Protoceratopsidae and Leptoceratopsidae are sometimes broken up and seen as a lineage leading to Ceratopsidae, but I'm tentatively retaining them.
    Ceratopsians generally get progressively larger; late ceratopsids are elephant-sized. Horns also are a common feature of ceratopsians; all ceratopsids and some more basal neoceratopsians possess at least prominent bumps over the nose and eyes. The shelf at the rear of the skull gets much larger, and in some cases becomes walled over. The holes in the frill, technically called fenestrae, both lightened the heavy frill and provided an attachment for jaw muscles, although this probably only occurred in the front sections of the frill. The main purpose of the frill was probably display; it was usually not strong enough to be much good in absorbing a predator's bite. To support the frill, the first three cervicals fused, as is seen in all ceratopsids.
    Ceratopsians evolved from bipeds, but become quadrupeds as they progressed. The position of the forelimbs has long been in dispute, but a consensus is forming that the arms were normally flexed at the elbow and slightly bowed out at the shoulder, with the hands facing to the outside.
    It is fairly well accepted that ceratopsians were often gregarious animals, in the sense that they grouped together. This is shown by bonebeds consisting of large numbers of bones belonging to one species. Whether or not they herded, in the sense of a complex social order, is (and probably always will be) unknown, but they certainly possessed much of the equipment to behave in such a manner: prominent display surfaces (horns, frills, and in some animals, tails) and what is best interpreted as sexual dimorphism of these features, such as prominent tall frills versus lower frills. Some of the postcranial material may also show dimorphic features.
    Ceratopsian jaws were very powerful, but not very good at grinding, working like giant shears. Ceratopsians probably sliced off short lengths of plants and swallowed them, letting the guts do the rest of the work. Basal ceratopsians like the psittacosaurids and leptoceratopsids may have been omnivorous.
    Ceratopsia as a whole is one of the best-known dinosaur groups, represented by hundreds of skulls and skeletons.

          |    |--Chaoyangsaurus
          |    `--Xuanhuaceratops
                              |     `--Yamaceratops
                                   |     `--Helioceratops
                                        |   |--Asiaceratops
                                        |   `--+--Cerasinops
                                        |        `--+--Montanoceratops
                                        |             `--+-Prenoceratops
                                        |                  `--+--+--Leptoceratops
                                        |                       |     `--Udanoceratops
                                        |                       `--+--Zhuchengceratops
                                        |                            `--+--Gryphoceratops
                                        |                                 `--Unescoceratops
                                                  |    |--Protoceratops
                                                  |    `--+--Ajkaceratops
                                        |         `--Bagaceratops

Ceratopsia: An entry that suggested ceratopsians in the late Campanian-early Maastrichtian (LK) of France turned out to be based on a typo for "ceratosaurians", but since we're starting to get European ceratopsians, it may just be a matter of time before it's right. 

Taxon or Taxa: Time/Place: Comments:
Yinlong downsi Xu X., Forster, Clark, and Mo J., 2006 Oxfordian (LJ) of China Yinlong is a basal marginocephalian/ceratopsian known from a nearly complete skeleton, its name a reference to the movie "Crouching Tiger, Hidden Dragon" (Yin Long means Hiding Dragon), which was filmed near where it was found. To picture it, think of a psittacosaur with a somewhat turtle-like skull. It helps to tie the pachies to ceratopsians, and may drag the heterodontosaurids in too.
Psittacosaurus: Osborn, 1923 (including Hongshanosaurus You, Xu, and Wang, 2003) P. mongoliensis (type) Osborn, 1923 Barremian-Aptian (EK) of China and Mongolia One of the best-known dinosaurs, Psittacosaurus is a small (less than two meters in length) biped with a distinctive skull profile; it has a prominent parrot-like beak and high nostrils, giving this animal a skull that looks almost squashed. Processing of food was accomplished with a combination of vertical and horizontal movements of the jaws, although not by using any joints besides the hinge of the mandible. Fittingly, the movement was parrot-like, and with the presence of gastroliths suggests they were eating nuts. The gastroliths have also been interpreted as ballast for a semi-aquatic animal. Several lines of evidence have been put forward for this, although it's mostly qualitative assessments of things like the apparent range of leg and arm motion, or a long and deep tail (which doesn't look unusually long and deep when compared to, say, hypsils). It doesn't hurt to put the idea out there, though.
20 species have been proposed, but not all are considered valid; however, it has the most well-supported species of any classic dinosaur. It has only four fingers, while ceratopsids have five; thus, it's not in the line leading directly to ceratopsids, due to its modified hand.
It was originally classified as an ornithopod (one specimen was named Protiguanodon). At one time, it was also thought to have something to do with ankylosaurians, due to what was interpreted as small armor studs on the body. Since then, the rostral bone at the front of the upper jaw has been described for what it is, and this creature classified as a very basal ceratopsian. Well over a hundred specimens are known for this animal; unfortunately, few mounts of it are to be seen in the United States. Pigeon-sized juvenile remains are known; in one case, 34 juveniles were found ("associated adult" turned out to be a Quaternary throw-in of a partially grown individual), together in an area with dimensions of 0.5 square meters. The sheer number in that space suggests some kind of grouping, as simple as a nest or as complex as a family gathering.
Psittacosaurus is known to have typical dinosaurian scalation; new specimens from the Yixian also indicate that at least some species had unusual bristles along the midline of the proximal third of the tail. These bristles appear to be about 15 cm long, and may have had a defensive function.
Recently, a new specimen of the opossum-sized EK Chinese triconodont mammal Repenomamus robustus was described, having been found to include partial remains of a juvenile Psittacosaurus in its gut region. The juvenile had a head-body length of about 14 cm, or about a third of its consumer's head-body length, and the fact that some of its long bones were articulated suggests that it was eaten in chunks. This find establishes that early mammals ate dinosaurs, although whether this was a case of predation or scavenging can't be established.
Psittacosaurus sibiricus has some features that seem quite unusual for Psittacosaurus, including a frill that is 15-18% of the skull length, three postorbital horns, a long shallow predentary, and two more presacral vertebrae (23 total) than any other known species of Psittacosaurus. It is known from at least two complete skeletons, and was described as the largest and most derived species of the genus. Photos show an animal that appears to have been the result of a breeding experiment with a basal neoceratopsian and a Psittacosaurus, followed by terrible acne or something. It had the typical psittac jugal horns, but also eyebrow nodes, nodes beside the eyes, and triangles flaring out just behind the nostrils. A new description found it to be closest to P. sinensis among psittacosaurids, and included it within Psittacosaurus, but noted numerous differences. Thanks to Frederik Spindler for pressing me about this strange animal, and providing me images of the skull!
P. houi (originally Hongshanosaurus), P. lujiatuensis, and P. major are probably all different growth stages/taphonomic accidents (i.e. crushing)/other variants of the same thing. The authors who published this preferred to use P. lujiatuensis, because Hongshanosaurus is based on a juvenile, but if they can all be considered the same, the oldest name ought to be used.
P. gobiensis Sereno, Zhao X., and Tan L., 2009 Aptian (EK) of China
P. houi (originally Hongshanosaurus houi You, Xu, and Wang, 2003) (includes P. lujiatunensis Zhou C.-F., Gao K.-Q., Fox, and Chen S.-H., 2006, and P. major Sereno, Zhao X.-J, Brown, and Tan L., 2007) Barremian-early Aptian (EK) of China
P. meileyingensis Sereno, Zhao, Cheng, and Rao, 1988 late Aptian or late Barremian (EK) of China
P. neimongoliensis Russell and Zhao, 1996 ?Barremian (EK) of China
P. sibiricus Voronkevich and Averianov vide Leschinskiy, Voronkevich, Maschenko, and Averianov, 2000 Aptian-Albian (EK) of Russia (western Siberia)
P. sinensis Yang, 1958 Aptian-Albian (EK) of China
P. xinjiangensis Sereno and Zhao, 1988 ?Valanginian-Albian (EK) of China

Chaoyangsauridae: Both genera here are mostly known for having been described informally or mentioned back in the mid 1980s, with somewhat different names. Both were considered to be either basal pachycephalosaurs, psittacosaurid-like basal ceratopsians, or otherwise right around that split.

Taxon or Taxa: Time/Place: Comments:
Chaoyangsaurus youngi Zhao, Cheng, and Xu, 1999 Tithonian (LJ) of China Long known under the informal names "Chaoyoungosaurus" and "Chaoyangosaurus" as an animal possibly more basal than the ceratopsian-pachycephalosaurian split, this animal has turned out to be a very basal ceratopsian, more primitive than Psittacosaurus. It is known from a skull, jaws, seven cervicals, a humerus, and a scapula.
Xuanhuaceratops niei Zhao X., Cheng Z., Xu X. and Makovicky, 2006 LJ of China Known as "Xuanhuasaurus" for twenty years, Xuanhuaceratops finally is published and comes out right where it was always said to be. It is based on four fragmentary specimens. Remains include two incomplete skulls, verts, a scapulocoracoid, a humerus, and an ischium. 

Ceratopsia i.s.:

Taxon or Taxa: Time/Place: Comments:
"Luanpingosaurus jingshanensis" (N.N.) Cheng vide Chen, 1996 (?Psittacosaurus) EK of China At this point your guess is as good as mine. There are reports that it is a synonym of Psittacosaurus.
"Psittacosaurus": "P." mazongshanensis (N.D.?) Xu, 1997 Barremian (EK) of China Nothing to see here, just dubious or potentially dubious species of Psittacosaurus.
"P." ordosensis (N.D.?) Russell and Zhao, 1996 (?P. sinensis) ?Barremian (EK) of China
"P." sattayaraki (N.D.) Buffetaut and Suteethorn, 1992 Aptian-Albian (EK) of Thailand

Neoceratopsia: Basal neoceratopsians are coming out of the woodwork, and it will be very interesting to see how they all sort out. For much of the 20th century, it was just Protoceratops and a couple of friends, but now there are a lot of good remains from the middle of the Cretaceous through to the end, from North America and especially Asia (and not just Mongolia and China, either). A few small internal lineages may become apparent, like Leptoceratopsidae.
    An unusual feature of all neoceratopsians is the fusion of the first three cervicals into a solid unit. It is believed that this was probably an adaptation to support the increasingly large head. The skull is perfectly balanced with a ball and socket joint at the end of this complex, giving it great mobility.
    An undescribed basal neoceratopsian is known from hips and caudals from the mid-late Albian (EK) of Idaho.

Taxon or Taxa: Time/Place: Comments:
Liaoceratops yanzigouensis Xu, Makovicky, Wang, Norell, and You, 2002 early Aptian (EK) of China A Yixian ceratopsian known from two complete skulls (a subadult and a juvenile) equipped with premaxillary teeth and a rudimentary frill, this small (described as "dog-size") animal is one of the most basal known neoceratopsians, if not the most basal neoceratopsian, more derived than Psittacosaurus but similar. 
Auroraceratops rugosus You, H., D. Li, Q. Ji, Lamanna, and Dodson, 2005  Aptian-Albian (EK) of China This basal neoceratopsian is based on a skull and lower jaw. Unusual characters include broad nasals, "fungiform" lacrimals (which means they have a sort of mushroom shape, bulging out), and rugosities on the jugals and lower jaw (hence the species name, I would assume). It was an approximate contemporary of Archaeoceratops. The authors found it to be the sister to the Coronosauria, a group which I don't have labeled here due to the recent inflation of basal ceratopsians, but which for simplicity's sake you can consider to be the node where Protoceratopsidae and Ceratopsoidea meet.
Aquilops americanus Farke, Maxwell, Cifelli, and Wedel, 2014 middle? Albian (EK) of Montana Aquilops is the first and to date only basal ceratopsian known from more than teeth from the Early Cretaceous of North America. It is based on a skull with that kind of vaguely turtle-like aspect that some other basal ceratopsians of similar grade have (it's that beak and that short square "frill" and those big round eyes that do it). The beak is strongly hooked, and there is a small boss on the anterior margin of the rostral bone.
Yamaceratops dorngobiensis Makovicky and Norell, 2006 ?Santonian-Campanian (LK) of Mongolia This basal neoceratopsian includes skull remains with a mix of basal and derived (mostly in the cheek and lower jaw) features, and a frill similar to those of Liaoceratops and Leptoceratops, this similarity suggesting that some function outside of display may be also present.
Archaeoceratops: Dong and Azuma, 1997 A. oshimai (type) Dong and Azuma, 1997 Barremian (EK) of China Archaeoceratops is known from a partial skeleton and skull. The Protoceratops-like teeth in the upper jaws contrast with the Psittacosaurus-like teeth in the lower jaws, suggesting this animal was less derived than the protoceratopsids. It was a small, fast animal.
A. yujingziensis is based on another partial skull, with a few verts, a partial scapula, the femora, and a few foot bones. It differs from the type species in some details of the maxilla and teeth.
A. yujingziensis H.-L. You, Tanoue, and Dodson, 2010 Aptian-Albian (EK) of China
Helioceratops brachygnathus Jin L., Chen J., Zan S., and Godefroit, 2009 late EK or early LK of China Helioceratops is a Auroraceratops/Yamaceratops-grade basal neoceratopsian. All of its distinguishing characteristics appear to be confined to the lower jaw; perhaps that's all that's known? The dentary is described as deep, and from the species name I'm judging that the jaw is short.
Koreaceratops hwaseongensis Lee, Ryan, and Kobayashi, 2010 Albian (EK) of South Korea Koreaceratops is known from most of the tail, part of the legs below the knees, and ischia. Probably more was there originally; the type specimen was found in a block incorporated into a dam. The genus is notable for its exaggerated caudal neural spines, which in some cases are five times longer than the underlying centra.

Neoceratopsia i.s.:  

Taxon or Taxa: Time/Place: Comments:
?Craspedodon lonzeensis (?N.D.) Dollo, 1883 Santonian (LK) of Belgium Craspedodon is based on teeth that were initially described as similar to those of Iguanodon. However, restudy suggested that the teeth were actually from (drumroll) a neoceratopsian (don't look smug; neoceratopsian teeth and iguanodontian teeth have a fair amount in common), possibly closest to (drumroll) ceratopsoids, which would make it the first neoceratopsian known from Europe.
Kulceratops kulensis (N.D.) Nesov, 1995 late Albian (EK) of central Asia This obscure animal is a basal neoceratopsian. It is based on a dentary fragment.
Microceratus gobiensis (N.D.) Mateus, 2008 (originally Microceratops gobiensis Bohlin, 1953) ?mid-late Campanian (LK) of China As the name suggests, this is a very small animal, less than a meter in length, although much of the remains are juvenile. Its proportions suggest it was one of the fastest neoceratopsians. Some scientists have suggested that its remains are indeterminate, though; the best material has been referred to Graciliceratops. The original name, Microceratops, is preoccupied by a hymenopteran (wasps, bees, and ants; Seyrig, 1952), necessitating the name change.

Leptoceratopsidae: While North America is not known for its basal neoceratopsians, leptoceratopsids seem to have caught on famously.

Taxon or Taxa: Time/Place: Comments:
Asiaceratops salsopaludalis Nesov, Kaznyshkina, and Cherepanov, 1989 late Albian (EK) or Cenomanian (LK) of Uzbekistan This is a small, poorly known neoceratopsian. It is known from parts of a skull and a phalanx (either a toe or finger bone). In recent studies, it tends to bounce around the base of Neoceratopsia.
Cerasinops hodgskissi Chinnery and Horner, 2007 early Campanian (LK) of Montana The specimens now known as Cerasinops have been kicking around for years, the type having been known as "Cera" since the 1980s, and the two referred specimens being described by Charles Gilmore as Leptoceratops sp. in 1939. It is known from most of the skeleton and skull. It shows a mix of characters known from Asian and North American ceratopsians, such as premaxillary teeth (Asian) and form of tooth wear (North American). Its forelimbs are very short (moreso than some psittacosaurids) and relatively gracile (as well as histologically distinct compared to load-bearing bones), indicating a biped. The type individual was at most 11 years old at death.
Montanoceratops cerorhynchus Sternberg, 1951 (originally Leptoceratops cerorhynchus Brown and Schlaikjer, 1942) latest Campanian-early Maastrichtian (LK) of Montana Displaying a taller nose horn (but see below) than its cousin Protoceratops, Montanoceratops also possesses a fin-tail. It is known from two specimens, including a partial skull. The nasal horn thought to distinguish it is actually a misplaced jugal (outwardly projecting in ceratopsians and responsible for the so-called additional "horns" of Pentaceratops).
Prenoceratops pieganensis Chinney, 2004 middle-late Campanian (LK) of Montana Prenoceratops is a new basal neoceratopsian, known from juvenile bonebed remains covering most of the skeleton (although only the skull has been described to date). It seems to be close to Leptoceratops, although with a longer frill and a longer, lower skull overall. It comes from the Two Medicine Formation of Montana. Unnamed material referable to the genus is known from the Oldman Formation in Alberta, of roughly comparable age.
Leptoceratops gracilis Brown, 1914 late Maastrichtian (LK) of Alberta, Montana, and Wyoming Seemingly one of the most basal neoceratopsians, Leptoceratops is also ironically one of the last. It is known from several good specimens. Although the type species is Lancian, there is new juvenile bonebed material that is Judithian in age which probably represents a new species (this doesn't seem to be the same material as that referred to Prenoceratops; basal neoceratopsians are an understudied part of the North American LK fauna, and Leptoceratops has long been the gold standard for comparison among what is starting to look like a NA radiation of closely-related critters).
Udanoceratops tschizhovi Kurzanov, 1992 late Campanian (LK) of Mongolia Based on a partial skeleton and skull, this is, at over four meters in length, the largest known bipedal neoceratopsian. It has an unusually deep low jaw. It appears to be closest to Leptoceratops.
Zhuchengceratops inexpectatus Xu X., Wang K., Zhao X., Sullivan, and Chen S., 2010 ?middle-late Campanian (LK) of China Zhuchengceratops is a derived Chinese leptoceratopsid, known from a partial skull and jaws, hyoids, 14 presacrals, ribs, and fragments. This specimen was found in a Shantungosaurus bonebed. It also shows more derived ceratopsian features, indicating convergences.
Gryphoceratops morrisoni Ryan, Evans, Currie, Brown, and Brinkman, 2012 late Santonian (LK) of Alberta Gryphoceratops (briefly Gryphognathus) is a derived yet early leptoceratopsid known from a dentary.
Unescoceratops koppelhusae Ryan, Evans, Currie, Brown, and Brinkman, 2012 late Campanian (LK) of Alberta Unescoceratops, like Gryphoceratops, is also a derived leptoceratopsid known from a dentary, although not as early.

Leptoceratopsidae i.s.:

Taxon or Taxa: Time/Place: Comments:
?Bainoceratops efremovi Tereschenko and Alifanov, 2003 late Campanian (LK) of Mongolia Based on a vertebral column, Bainoceratops may be affiliated with Leptoceratops and Udanoceratops, or it might be a synonym of Protoceratops.

Protoceratopsidae and friends: After having come under attack as a paraphyletic family throughout much of the 1990s, it now appears that a small cluster of dinosaurs may make up a true Protoceratopsidae.

Taxon or Taxa: Time/Place: Comments:
?Graciliceratops mongoliensis Sereno, 2000 ?Santonian (LK) of Mongolia Graciliceratops is based on possibly juvenile remains (partial articulated skull and skeleton) once referred to Microceratus.
Bagaceratops rozhdestvenskyi Maryanska and Osmolska, 1975 (including: Breviceratops kozlowskii Kurzanov, 1990 [originally Protoceratops kozlowskii Maryanska and Osmolska, 1975]; Gobiceratops minutus Alifanov, 2008; Lamaceratops tereschenkoi Alifanov, 2003; Platyceratops tatarinovi Alifanov, 2003) (?Magnirostris) ?mid Campanian (LK) of Mongolia Known from remains of over twenty individuals, Bagaceratops is one of the best known basal neoceratopsians. Its skull has a small node over the nose. It is sometimes assigned to Bagaceratopidae (note spelling), along with some of its synonyms.
Now assigned here are the partial juvenile remains first named Protoceratops kozlowskii and later called Breviceratops; the features thought to make it different from Bagaceratops have turned out to be growth-related.
Also probably representing growth stages of Bagaceratops are Gobiceratops, Lamaceratops, Magnirostris, and Platyceratops, based on skulls of various-sized basal neoceratopsians. They may not necessarily all be synonyms, either at the species or genus levels, but having several of pretty much the same thing at somewhat different sizes running around in a single time and place is unlikely. I've got Magnirostris separate for now because of some fairly distinct cranial features and its different temporal and geographic placements.
Ajkaceratops kozmai Ősi, Butler, and Weishampel, 2010 Santonian (LK) of Hungary Ajkaceratops is the first ceratopsian described from Europe that is known from more than teeth (well, except possibly Stenopelix, but that guy switches affiliation every few years or so). It's known from the upper and lower beaks of something similar to Bagaceratops and Magnirostris.
Protoceratops: Granger and Gregory, 1923 P. andrewsi (type) Granger and Gregory, 1923 late Campanian (LK) of Mongolia and China Known from remains belonging to at least eighty individuals of a wide range of ages and what appears to be both sexes, Protoceratops is one of the most famous dinosaurs. Skulls interpreted as coming from males have strongly-arching nasal regions and tall frills, while females have less arched nasal regions and lower, smaller frills. Protoceratops also has a "fin-tail" composed of tall caudal spines. It had long been considered to be the dinosaur responsible for many nests in Mongolia, but it now appears at least some of the nests and eggs actually belong to an oviraptorid, once accused of seeking out Protoceratops eggs to eat.
The large eyes of Protoceratops are consistent with nocturnal habits, which would have been useful in an arid setting. If it was, it needed to be somewhere safe during the day, and there are not a few examples of Protoceratops found buried "standing up", like they were in holes...
P. hellenikorhinus Lambert, Godefroit, Shang, and Dong, 2001 late Campanian (LK) of China New species P. hellenikorhinus, known from at least nine individuals, was larger than the type, and had a vaguely chasmsosaurine-like frill, larger jugal horns, and double nasal horns (actually more like one nasal horn with the top split into two parallel ridges).

Protoceratopsidae i.s.:

Taxon or Taxa: Time/Place: Comments:
Magnirostris dodsoni You and Dong, 2003 (?Bagaceratops) late Campanian (LK) of China Another possible "bagaceratopid" (or example of Bagaceratops), this one has an unusually large rostral, giving it a big beak. It is based on a skull missing the frill, but with nubbins of brow horns, and a well-developed nasal ridge.


Taxon or Taxa: Time/Place: Comments:
Zuniceratops christopheri Wolfe and Kirkland, 1998 mid Turonian (LK) of New Mexico Temporarily the last name on the list of valid dinosaur names (now supplanted by Zupaysaurus), Zuniceratops is based on partial remains that suggest it is closest to basal ceratopsids, with brow horns and all (although no nasal horn). The frill is fenestrated, but lacks epoccipitals. Interestingly, the teeth appear to be single-rooted, unlike the double-rooted teeth of true ceratopsids. A bone thought to be the squamosal has turned out to be a therizinosaurian ischium (Nothronychus). Remains from several individuals are known.
New bonebed material from at least five individuals provides evidence that the teeth became double-rooted with age, and the brow horns also grew substantially with age.
Turanoceratops tardabilis Nesov, Kaznyshkina, and Cherepanov, 1989 Turonian (LK) of Kazakhstan Turanoceratops has bounced around a bit over its history in the nebulous space betwixt basal ceratopsid and the ceratopsians just outside. Restudy indicates that it was more derived than Zuniceratops, but also had the "two brow horns and no nasal horn" look going.

Ceratopsidae i.s.: Ceratopsids are distinguished from the more basal neoceratopsians by their enlarged frills and full-blown horn cores. They are also much larger and are in no way bipedal.
    A couple of decades ago some researchers suggested that ceratopsids and their frilled ancestors had their frills completely encased in flesh and muscle. This suggestion led to some rather odd illustrations. Since frills tend to have distinctive ornamented margins, it is more logical to suppose the frills were not encased in muscle. For example, why would these animals have ornamented margins to their frills if they were just going to be covered?  If they truly were encased, all ceratopsids should have roughly the same form of frill margins.
    Three new ceratopsid skulls are known from the lower Campanian (LK) of Utah.

Taxon or Taxa: Time/Place: Comments:
Agathaumas sylvestris (N.D.) Cope, 1872 (?Torosaurus or Triceratops) late Maastrichtian (LK) of Wyoming Based on a section of the hip and back, this historically significant animal (first named ceratopsid) is just a large indeterminate ceratopsid, probably either Triceratops or Torosaurus.
Ceratops montanus (N.D.) Marsh, 1888 middle-late Campanian (LK) of Montana Historically significant as the namesake of Ceratopsidae, this animal is very poorly known. Its type consists of horn cores and skull fragments, which are similar to both Chasmosaurus and Avaceratops bones. New material may belong to it. With the discovery of centrosaurines with long brow horns, like Albertaceratops, it's no longer safe to assume that Ceratops was a "ceratopsine".
(N.D.) Cope, 1876
D. encaustatus (N.D.) (type) Cope, 1876 middle-late Campanian (LK) of Montana These names, except for D. peiganus, are all based on typical ceratopsid teeth, with split roots (with some hadrosaurid teeth mixed in for D. encaustus), but beyond that can give little useful information. D. peiganus is represented by a tooth from the anterior end of the dentary tooth row, which looks more like an ankylosaurian or hypsilophodont tooth than standard-issue ceratopsid.
D. bicarinatus (N.D.) Cope, 1876
D. haydenianus (N.D.) Cope, 1876
D. peiganus (N.D.) Cope, 1876
"Monoclonius": "M." fissus (N.D.) Cope, 1889 middle-late Campanian (LK) of Montana This is just ceratopsid junk.
"M." recurvicornis (N.D.) Cope, 1889 This animal is known from the upper part of a ceratopsid's face that looks a bit like it had the equivalent of the virus that gives rabbits weird horn-like head tumors. There is a long-based short procurved blunt nasal horn, a couple of knobby brow horns, and some other knobs.
"Triceratops" maximus (N.D.) Brown, 1933 late Maastrichtian (LK) of Montana The remains assigned here can only be classified as that of a large ceratopsid.

Centrosaurinae and Chasmosaurinae: These two subgroups are based on horn arrangement and relative extent of the squamosal. The two squamosal bones make up a side of the frill each, flanking the parietal bone which makes up the bulk of the frill (and usually encloses two fenestrae, one on either side of the frill midline). In centrosaurines, the squamosals are short and never reach the end of the frill. In chasmosaurines, the squamosals are long and reach the end. Both groups have bones called epoccipitals, which often line the edges of the frill and give it a jagged look.
    These bony projections were probably only for show. Centrosaurines also have a flange of bone pointing to the front of the nasal fenestra from the rear, and have nasal horns which are more prominent than brow horns. Chasmosaurines have a flange in the nasal fenestra pointing back from the front, and more prominent brow horns. Both groups have very similar postcrania, and the best way to separate taxa is to use adult skulls. One very interesting possibility is that, if they were social animals, perhaps the presence of dominance hierarchies repressed full expression of these features for most individuals, as happens in modern social animals.
    Some ceratopsids have squamosal fenestrae, which have puzzled researchers for a long time. It may be that the chasmosaurine versions are natural, while the centrosaurine versions are lesions, remnants of puncture wounds (like, say, from another centrosaurine's nasal horn). 

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