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This is a complex question. Deciding if a fish is asleep depends on many factors, one of which is the definition of sleep.
In humans, the transition to sleep (or even closing the eyes) involves specified changes in the pattern of brain waves in the area of the brain called the neocortex. Fishes don't have the same degree of neocortical development as mammals and thus don't display these brain-wave patterns associated with sleep. So, as far as brain-wave patterns go, fishes don't sleep.
If, however, you define sleep as a combination of a reduced metabolic rate, slowed physical activity, lowered response to stimuli and the assumption of a resting posture, then many fishes do sleep. Perhaps the best-known 'sleepers' are the parrotfishes (family Scaridae). Many parrotfishes find a suitable spot on the seafloor and secrete a mucus envelope in which they spend the night.
Such fishes normally swim away from a diver by day, but can easily be approached and even gently handled at night. Many fishes, however, seem not to sleep. Pelagic species such as tunas and some sharks never stop swimming. One theory suggests that during sleep, sensory information (predominantly visual) gathered during the day is processed to form memories. Fishes that swim constantly in blue oceanic waters receive little 'unusual' visual input and require less 'memory-processing time' and thus need no sleep. This is supported by studies on several species of blind fishes that live in caves. These sightless fishes do not sleep.
Having said all that, though, I did enjoy the simple answer of a young neighbour who told me with confidence that fish can't sleep because they don't have eyelids.
The text on this webpage is from McGrouther, M.A. 2003. Do Fishes Sleep? Nature Australia. Spring. 27(10):82.