(Dungeons & Dragons)
"Elves sleep with their eyes open..."
The elves typically do not sleep as such, and need far less time per day to rest. While your average human sleeps for 8 hours a night, elves get by with 4 hours of rest. This covers only the mental necessity of rest - physical exhaustion may still require a longer rest.
The elven rest, called the reverie, is unlike human sleep and far more resembles meditation. In reverie the elf falls into a deep trance-like state, completely closing out the outside world and revisiting and reliving their own memories of past events. As in dreaming, the elf has no control which memories will surface, so this will include both bad and good memories.
Interestingly, it seems not all elves can or want to experience reverie. Elves can sleep instead of falling into reverie - whether or not the sleep involves dreams is another debated matter. Of particular note of elves are the drow, a subrace that is getting completely alienated from other elves in many ways already. My sources (heated debate forums and web pages) conflict with this, but some say the drow have lost their capability of falling into reverie due to their curse from the Seldarine. Well, my copy of PHB and FRCS seem to disagree and make no special mention of this (PHB says elves fall in trance, FRCS doesn't deny this and says drow have all elf traits except for the listed); Besides, I remember reading from some Forgotten Realms novel that most of the drow still fall in reverie like surface elves, but increasing part of the younger drow seem to either prefer "normal" sleep, or are actually incapable of falling into reverie - this may be due to the fact that drow have such miserable lives that revisiting the memories is getting more and more annoying.
It should should be noted that the 3rd edition D&D Player's Handbook refers to the reverie simply as the "trance".