Here's a quick breakdown of their largest exhibits.
Small/ or desperate need of new residents
- African wild cat
- Blue bellied roller
- Blacksmith Plover, Chestnut Weaver
- Bat-eared fox
- Red-billed Hornbill
- Ocelot (debatable)
- Bobcat
- Swift Fox
Thanks to @Moebelle for an excellent summary of the Desert Dome's largest exhibits, although it must be pointed out that there is a fantastic selection of reptiles in the building (particularly some choice rarities from Australia) and those exhibits are mainly all rather decent. I think that @nczoofan has a good point in that the Desert Dome looks incredible and everyone is in agreement with that statement. However, I absolutely concur that "most of the mammal exhibits seem to be on the small side". I think that the overall quality of the Desert Dome would be immeasurably improved by the switching in of half-a-dozen smaller species in place of larger species.
I'm not quite sure why Omaha hasn't addressed more of its failings. Anyone who visits cannot fail to be genuinely impressed with the size and scale of so many of the animal habitats. All of the new stuff, particularly African Grasslands and Asian Highlands, is right up there with the best zoos anywhere. Many of the older buildings are still superb, it's a full-day zoo, and I think that with every mega-budget addition it gets harder and harder to lay claim that San Diego is America's 'best' zoo. However, why hasn't Omaha tweaked some of the mammal enclosures in the Desert Dome or Lied Jungle? Does the public really need to see bobcats and African wild cats in tiny, all-indoor exhibits? Would visitors care if the Malayan tapirs were shifted out of the Lied Jungle and in their place were otters or a type of crocodilian instead? As great as Omaha is, and it is often hailed as one of the very best zoos on the planet, there are clearly shortcomings in some of the buildings. My biggest complaint would probably be Expedition Madagascar, which opened in 2010 and yet has lemurs entirely indoors with zero outdoor access for some species.
To be fair to Omaha, they have consistently been addressing these issues, though perhaps not as rapidly as they could. First time I saw the Lied Jungle there were Clouded Leopards and Asian Golden Cats in two small caves. Similarly, the Desert Dome started out with Leopard and Puma in background cliffs as well as the aforementioned hyrax and wallaby. The thing I find most odd are the Bobcat and Swift Fox. Both are native to Nebraska and would do just fine outside all year.