The Generalife is on a nearby hill and was conceived as a recreation area for the Nazarite court. Its gardens, fountains, water tanks, ponds, etc. are outstanding. It was built in about 1319 by Aben Walid Ismail.
Life other parts of the Alhambra, the Generalife has a history of Kings, Queens and Princes. Perhaps one of the more fascinating stories is that of Zoraya, the harem favourite of the emir Abu Al-Hasan.
If you remember me mentioning the emir Abu Al-Hasan (and Zoraya) from my overall Granada web page, he was the emir who ran afoul of his wife Alixa (possibly because of his relationship Zoraya) and that lead to a civil war with his son (which the Emir eventually lost, and ultimately lead to the "victor" (his son's) loss of Granada to the Christians Isabella and Fernando). Well apparently Abu Al-Hasan caught Zoraya with the leader of the Abencerraj clan in the Generalife, presumeably up to something that the emir did not approve of. This resulted in the murders, in the Palacio Nazaries, of many of the nobles of the Abencerraj clan (probably the emir's revenge). There may also have been other reasons for the muders, as the Abencerrajes also are purported to have favoured the rival Boabdil (the emir's son by Alixa), in the palace power struggle.
Washington Irving tells an interesting account of the Abu Al-Hasan, Zoraya, and much of the politics that took place in this ancient kingdom: Click here for a link to Washington Irving's account (from his "Chronical of the Conquest of Granada".
I suspect there are many more stories about the Generalife, and indeed of the Alhambra, if there were only more records to tell of it