The famous Herod Atticus Odeon dominates the western end on the south slope of the Acropolis. It was the third Odeon constructed in ancient Athens after the Pericles Odeon on the south slope (fifth century) and the Agrippas Odeon in the ancient Agora (15 BC). The construction of the monument during the second century AD was sponsored by Tiberius Claudius Herod Atticus, renowned offspring of an important Athenian family and a benefactor; he thus acted in remembrance of his wife Regilla, who died in 160 AD. The exact date of construction is unknown, but it was certainly built sometime after Regillas death and 174 AD, when the traveler and geographer Pausanias visited Athens and referred to the monument in great admiration. The roofed Odeon served mainly musical festivals, and could host up to 5,000 spectators. The Odeon was destroyed in 267 BC at the incursion of the Heruli (or Erils), who burnt and flattened many buildings in ancient Athens; it was never reconstructed contrary to other edifices that suffered damages. In later years, the Odeon was incorporated within the fortifications of the city of Athens.  The monument was restored during 1952-1953 using marble from the Dionysus area; since 1957 it stages art festivals (concerts, ancient drama performances, etc.) mostly in the framework of the Athens Festival.
http://greekfestival.gr/en/venues/view/odeon-of-herodes-atticus
Sources: DOMH, and Maria Kosma, archaeologist
http://odysseus.culture.gr/h/2/eh251.jsp?obj_id=6622
Google map from Nestorion (http://www.nestoriohotel.gr) to Odeon Herod Atticus: https://goo.gl/maps/CAlCj
Tickets: http://greekfestival.gr/en/content/page/tickets