Research has demonstrated that cats and dogs do, in fact, dream. Dr. William C. Dement conducted early research into dreams more than 50 years ago with the use of rudimentary electrodes that measured the electrical activity of the brain. These measurements identified the brain waves with an electroencephalogram (EEG). Eye movements that were seen in sleep could also be monitored with an electrooculogram (EOG). With careful observation, it was noted that rapid eye movement (REM) sleep was associated with a different pattern of brain activity. Sleep stages were thus divided into non-REM and REM sleep. Vivid dreams were more clearly associated with REM sleep.
These same patterns have been demonstrated in other species, including cats and dogs. During REM, while the body remains paralyzed so that dreams are not acted out, the eyes will still move. There may also be twitches of muscles. When the muscles are not properly paralyzed, REM behavior disorder (RBD) may result in humans. You might even observe movements of your pet’s legs during sleep, suggesting a similar failure to paralyze the body completely. A dog may even bark softly.
Though we may be able to demonstrate that REM sleep occurs in cats and dogs, we cannot be certain of what they dream. If they are like humans, the content of their dreams probably reflects their waking experiences, just as ours do.
Kryger, MH et al. "Principles and Practice of Sleep Medicine." ExpertConsult, 5th edition, 2011.