The Milinda Panha is a Buddhist text written more than 2,000 years ago. It takes the form of a dialogue between Indo-Greek King Menander I and a Buddhist sage.
At one point, the king is interested in the meaning of dreams. The sage talks about the possible influences of the waking life on dreams and then mentions that they occur in the period between falling asleep and deep sleep. This insight is fascinating, because scientists weren't able to make much more progress than this until the 1950s. For something so universal, it's odd that scientists really didn't start to understand what goes on when we sleep until relatively recently.
The breakthrough came when American physiologists Eugene Aserinsky and Nathaniel Kleitman were making observations of sleeping infants and recording different phases of sleep. In particular, they noticed that the eyes would sometimes jerk and twitch under the eyelids. The pair named this "rapid eye movement" (REM) sleep and later found that it didn't just occur in babies but in people of all ages. We've seen it since in all mammals and in birds, too.