The Garden of Gethsemane still exists! It is a grove of Jerusalem olive trees on the lower slopes of the Mount of Olives, across the Kidron Valley from the Old City walls in Jerusalem.
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The Garden of Gethsemane, the place where Jesus prayed and agonized about his passion, and where he was arrested after Judas betrayed him, sits outside of Jerusalem across a small valley. The Mount of Olives stretches as a ridgeline opposite the Temple; the garden was on this hillside (see it on a map here).