Is there REM sleep in reptiles? A key question, but still unanswered

•A third of the papers on sleep in reptiles reported a REM sleep-like state.•However, none of these studies provides enough clues to postulate a true homology with mammalian REM sleep.•As reptiles are not capable of expressing all the traits that define mammalian REM sleep, the quest to identify the same state in reptiles, could be unfounded.•The origin of REM sleep will be revealed through more comparative studies focused on how animals sleep, rather than whether they have REM sleep, as defined in mammals.