What's the oldest building in the world that is still in use (i.e. used for something other than a tourist spot).
Pantheon in Rome (126 AD).
Most of the older buildings in the Wiki list ( http://en.wikipedia.org/wiki/List_of_the_oldest_buildings_in_the_world ) are either not in use, or used as tombs only, or were reconstructed significantly.
The Epidaurus Theatre (ca. 300-340 BC), the Delphi theatre (4th century BC) and the Odeon of Herodes Atticus (161 AD) in the Acropolis of Athens (known locally as the the Herodeon), still fulfil their original purpose, all three are constantly used as venues for various festivals. The ancient theatre in Dion is also used occasionally.
The Colosseum (completed in 80 AD) could also qualify, while not in constant use as with the Pantheon that DVK already mentioned, it is used by the catholic church for the Via Crucis ceremony on Good Friday. Furthermore in July 2000 the National Theatre of Greece performed Oedipus Rex in the Colosseum.
Lastly, the remains of the Temple of Hera (590 BC, destroyed by an earthquake in the 4th century BC) in Ancient Olympia is the location where the torch of the Olympic flame for the Modern Olympics is lit. A continuous flame was maintained at the sanctuary of the temple during the Ancient Olympics, and the temple was also the location where the olive wreaths for the victors were displayed during the games.
The upper story of the Theater of Marcellus (ca 13 BC) in Rome is a block of apartments.
While not exactly a building, the Western Wall in Jerusalem ( http://en.wikipedia.org/wiki/Western_Wall ) is a site in which daily praying takes place. It was constructed around around 19 BCE.
The Roman theatre in Caesarea.
Stonehenge and other megalithic sites in Britain are still, or pehaps more accurately, once again used for religious purposes.