REM sleep is a period of sleep that is very deep, during which your eyes and muscles make many small movements. It is the period during which most of your dreams occur. REM is an abbreviation for `rapid eye movement'. The rapid, periodic, jerky movement of the eyes during certain stages of the sleep cycle when dreaming takes place. a US rock band popular in the 1980s and 1990s. Their albums include Automatic for the People and Monster