The place in modern Jerusalem claimed by a number of church denominations to be the Garden of Gethsemanè, where Jesus came to pray with his disciples, and where Judas betrayed Him. Modern scholarship acknowledges that the location of Gethsemane is unknown. Gethsemanè means "place of the olive press" and according to indications from the Gospels, Gethsemane was somewhere on or around the Mount of Olives - which is quite a big area. This supposed Gethsemanè is basically what is left of an olive grove, with some footpaths and recent attempts at gardening.
The olive trees are said to be left over and recovered after the Romans burned down the groves in 70 AD. Recent carbo dating on the trees in 2012, found that the oldest sample dated from 1092 AD.