From "The Illustrated Cat's Life" by Warren Eckstein and Fay Eckstein:

Do cats dream?

This is still one of the great unanswered questions.

Most owners have long had sneaking suspicions about feline sleeptime imagery as they watch paws that twitch, legs that run and whiskers which flutter wildly.

But scientists are only now confirming that the odds that cats do really dream are indeed quite good.

Some recent studies seem to indicate that cats enter into REM (rapid eye movement) sleep, the type of sleep during which, in humans, dreams generally take place.

-- Robin Palmer