I'm pretty sure that cats have REM sleep - At least mine do!
Not all animals experience REM sleep, but some do. The amount of REM sleep an animal gets seems to be related to its safety in its environment. The more danger for the animal, the less REM sleep. The amount of REM sleep is also related to how developed the animal is at birth. Animals born fairly mature, such as sheep (whose REM Sleep is only 1% of their daily sleep), have less REM sleep than animals that continue to grow and develop more after they are born, such as humans.
Amphibians and Fish have never been found to have any similar experience to REM sleep (1).
Birds spend about 5% of their sleep time in REM sleep. They seem to have very brief dreaming episodes. (2)
Most Mammals experience REM sleep. Ruminating animals, animals that chew their cud, get very little (if any) REM sleep.
Not all animals that experience REM sleep dream. Some think that perhaps only mammals seeming to be more advanced in thinking than others dream during REM sleep. However, dolphins, thought to be highly intelligent do not have any REM sleep whatsoever! Dreaming mammals besides humans include: cats, dogs, chimpanzees, horses, cows.
Observing REM Sleep in a Cat or Dog
Cats and dogs have dreaming periods usually about 15 minutes into their sleep.(3) Their REM sleep is easy to observe. Simple indicators are movements under the eyelids, changes in breathing patterns, twitching of the ears or whiskers, or occasional spasms or jerks of the face. They may even make running motions with their paws. If someone placed food before a sleeping dog's nose, the dog may even start chewing.