The Panathenaic Stadium, known to locals as the Kallimarmaro ("beautifully marbled"), sits in the green park of Pangrati between Plaka and the National Garden. It is the only stadium in the world built entirely of white marble, and the only place where you can run on the track that hosted the first modern Olympic Games in 1896.
Pay the small admission fee, take the audio guide, and walk into the empty arena: it is a strange and impressive space, a U-shaped marble bowl seating 50,000 in unbroken stone rows. Climb to the top of the seats for a view back across central Athens to the Acropolis. The Olympic flame is still kept here.