What number of stages of sleep are there?

Four.: Sleep is divided into two general stages, REM (Rapid Eye Movement sleep), and NREM (non-REM) sleep. Furthermore, NREM sleep is currently divided into three stages (1,2 and 3), although up until recently there was defined a fourth stage as well. REM sleep is thought to be where most dreams occur. NREM sleep is staged by depth (stage 1 is drowsiness, 3 is deep sleep, 2 is in the middle).