The central theme of this paper deals with the phenomena of REM sleep in relation to dreaming. The physiological mechanisms of REM sleep, with primary characteristics of high brain activity and paradoxically low muscle tone, is explained. The function of REM sleep is discussed, particularly in terms of its cognitive and homeostatic functioning, but without being able to draw any final conclusion. Arguments are given to support the existence of a third state of consciousness, namely REM sleep, aside from the two commonly accepted states of ’wakefulness’ and ’slow wave’ sleep or non-REM sleep. REM sleep may not even belong to the domain of sleep itself, given the fact that REM sleep may occur during, or interfere with, wakefulness. Another topic discussed is the supposed relationship of REM sleep to dreaming. Based on a number of considerations, it is concluded that intense brain activity, regardless of its origin, may be the essential causal factor of dreaming. Because high brain activity always accompanies REM sleep, it looks as if there is a causal relationship between the two. This is refuted however by the occurrence of dreams during slow wave sleep, and also by the existence of hypnagogic hallucinations during naps or while falling into sleep.