It's a question that has long fascinated both scientists and pet parents alike. A 2001 study of rats by MIT researchers suggested that the animals not only dream but dream about past events in the same way that humans do. The researcher were able to take recordings of electrical impulses from the rats' hippocampus, the part of the brain that controls memory, while the rats were awake and match those impulses to nearly identical impulses that occurred while they were asleep. Because the electric impulses were so clear and consistent, the researchers were able to pinpoint the exact part of the maze that the rats were dreaming about.
Considering that dogs have a much more complex brain than rats, and their brains more closely mimic those of humans, it is reasonable to assume that dogs also dream.
Dr. Stanley Coren of the University of British Columbia suggests that you can physically see when a dog is dreaming, as the dog's eyes may move behind his closed eyelids while he is in a dream state. These eye movements are typical during REM sleep, the period of sleep in which humans dream, as the eyes view the dream images in the same way they view real images of the world during waking hours.