The modern perissodactyls are, sadly, but a shadow of their former glory. Once among the planet's dominant herbivores, the odd-toed hoofed mammals have become reduced to less than twenty living species (the majority of which are critically endangered to boot). Nevertheless, their secure position as charismatic megafauna means that they are familiar animals to most people (at least conceptually). One species in particular, the horse Equus caballus, has developed a close association with humanity and holds a high position in the human psyche (or at least the Eurasian and American psyche). But this post won't be dealing with horses - this is for the other side of the perissodactyls. The Ceratomorpha may not have been blessed with the aesthetic appeal of the horses, but they're not without their charms.
The Ceratomorpha contains two living families, the Tapiridae (tapirs) and Rhinocerotidae (rhinoceroses). Most of the people reading this will, I'm guessing, probably be familiar with the appearance of both, though rhinos do get given a little more press than tapirs*. Rhinos are also marginally more diverse in the modern environment, with five species to the tapirs' four - but considering that at least two of the rhino species are hovering on the brink of extinction, that may yet change. As regards fossil taxa, the limits of Ceratomorpha are a little more hazy, mostly because different authors have applied slightly different concepts for 'Ceratomorpha' vs. the related name 'Tapiromorpha'. In my opinion, the most sensible definitions for both are those proposed by Holbrook (1999), who used 'Ceratomorpha' for the crown clade of tapirs + rhinos, and 'Tapiromorpha' for the total group of anything more closely related to tapirs and rhinos than to horses. These definitions are better than the alternatives proposed by Froehlich (1999) in being agnostic as to whether the extinct chalicotheres are tapiromorphs (as supported by Froehlich, 1999, among others) or not (as indicated by, e.g., Hooker & Dashzeveg, 2004, who placed chalicotheres outside the perissodactyl crown group).
*Completely unrelated aside to everything else - the Japanese name for 'tapir' is 'baku'. Originally, a baku was a trunked mythical creature that was supposed to feed on people's dreams (particularly nightmares), but living tapirs have a somewhat more material diet. This isn't the only example in Japanese of a living exotic animal being granted the name of a pre-'existing' mythical creature: giraffes are known as 'kirin'.
Within the Ceratomorpha, then, the primary division is between the Tapiroidea and the Rhinocerotoidea, each including (obviously) the taxa more closely related to one of the living families than the other*. The Rhinocerotoidea are far better understood that the Tapiroidea, which have a more spotty fossil record. Four families are generally assigned to the Tapiroidea - Helaletidae, Lophialetidae, Deperetellidae and Tapiridae. The mostly Eocene 'Helaletidae', small tapiroids of North America and Eurasia, however, are probably paraphyletic with regard to other tapiroids. Holbrook (1999), for instance, excluded the North American genus Heptodon from the Helaletidae as he found it to be sister to all other tapiroids. At least one 'helaletid' genus, the North American Oligocene Colodon, is of interest because it possessed significantly retracted narial opening, indicating the presence of a trunk as in modern tapirs. However, Colbert (2005) included Colodon within the Tapiridae, closer to modern tapirs than previous authors.
*Be warned, though - older references tend to use the term 'tapiroid' as a grade concept for non-rhinocerotoid tapiromorphs, including a number of taxa that would be regarded as stem-Ceratomorpha in this post.
The Lophialetidae and Deperetellidae were two strictly Asian late Eocene families that suffer from a severe lack of study. Both have been mostly regarded as tapiroids for as long as they have been known, but Holbrook (2001) pointed out that the evidence for doing so is pretty slight. Members of both families (supported as forming a monophyletic clade by Holbrook, 1999) showed a reduction in the number of toes from four to three and a longer, more slender foot than other tapiroids, indicating that they were more cursorial (Radinsky, 1969). The Eocene also saw the appearance of the first Tapiridae, which (with the possible exception of Colodon, depending on its position) seem to have been the only tapiroids to survive into the Oligocene. The earliest tapirid genus, Protapirus, has been described from both North America and Eurasia, but its monophyly is uncertain (Colbert, 2005).
The other superfamily, Rhinocerotoidea, has had a much greater diversity described, both in terms of number of species and morphological range. Three major families have been recognised - the Amynodontidae, Hyracodontidae and Rhinocerotidae. The middle Eocene to Middle Oligocene Amynodontidae were the really unfortunate members of the superfamily appearance-wise - as the saying goes, they would have not only been hit with the ugly stick, they would have fallen out of the ugly tree hitting every ugly branch on the way down. Amynodontids have mostly been characterised as subaquatic, like modern hippos, but Wall (1998) points out that only a single derived subgroup, the Metamynodontini, shows adaptations for such a lifestyle. The remaining amynodontids would have been terrestrial. One group of amynodontids, the Cadurcodontini, appears to have convergently evolved a short trunk like that of the tapirids.
The Eocene to Oligocene Hyracodontidae are the real taxonomic dog's breakfast of the Rhinocerotoidea. Radinsky's (1966) influential definition of which taxa should be included in Hyracodontidae essentially amounted to "anything which doesn't belong to Amynodontidae or Rhinocerotidae". Characters that have been suggested to support a monophyletic Hyracodontidae, such as an elongate foot and three toes, are also found in other perissodactyls (in fact, if you look upwards you'll notice that I mentioned the exact same characters in connection with Lophialetidae). In general, hyracodontids were more cursorial than other rhinocerotoids, and small genera such as Hyracodon would have been fairly pony-like. Also usually included in the 'Hyracodontidae' where the gigantic Indricotheriinae, of which the central Asian Oligocene genus Paraceratherium is famed as the largest known land mammal*. The phylogenetic analysis of Holbrook (2001) supported treating the smaller Hyracodontinae as a separate family from the larger Indricotheriinae, but failed to support the latter as monophyletic.
*Though sometimes in disguise. Paraceratherium has held a few different names over the years - 'Indricotherium' and 'Baluchitherium' are two of the most commonly used. It remains a rather fraught question as to whether all these represent the same animal, or a number of closely related animals.
And finally, the Rhinocerotidae. But I think I've rabbited on and wasted enough of everyone's time for today, so I guess I'll have to leave the hippo-like Teleoceras, or the double-horned Menoceras (two horns side by side, that is), or the gigantic Elasmotherium (with what must have been one of the most terrifying pieces of headgear this side of Arsinoitherium) for another time.
Colbert, M. W. 2005. The facial skeleton of the Early Oligocene Colodon (Perissodactyla, Tapiroidea). Palaeontologia Electronica 8 (1): 8.1.12A.
Froehlich, D. J. 1999. Phylogenetic systematics of basal perissodactyls. Journal of Vertebrate Paleontology 19 (1): 140-159.
Holbrook, L. T. 1999. The phylogeny and classification of tapiromorph perissodactyls (Mammalia). Cladistics 15 (3): 331-350.
Holbrook, L. T. 2001. Comparative osteology of early Tertiary tapiromorphs (Mammalia, Perissodactyla). Zoological Journal of the Linnean Society 132 (1): 1-54.
Hooker, J. J., & D. Dashzeveg. 2004. The origin of chalicotheres (Perissodactyla, Mammalia). Palaeontology 47 (6): 1363-1386.
Radinsky, L. B. 1966. The families of the Rhinocerotoidea (Mammalia, Perissodactyla). Journal of Mammalogy 47 (4): 631-639.
Radinsky, L. B. 1969. The early evolution of the Perissodactyla. Evolution 23 (2): 308-328.
Wall, W. P. 1998. Amynodontidae. In Evolution of Tertiary Mammals of North America: Terrestrial Carnivores, Ungulates, and Ungulatelike Mammals (L. L. Jacobs & K. M. Scott, eds.) pp. 583-588. Cambridge University Press.