Before the concert at Aya Irini, Eva and I met up at the Hippodrome. Despite the fact that every time I go there some tout will try to chat me up, I just love it. I think it's because of the history of the place and that you can almost visualize what it used to be. And despite the random guys hanging around, it's also really quiet when compared to the touristy part of Sultanahmet.
According to Wikipedia, Emperor Septimius Severus built the hippodrome here in the year 203, no doubt to give the citizens of Byzantium some entertainment. (A hippodrome was a long oval-shaped arena where chariot races and other events were held.) Constantine enlarged it in the fourth century, and it became the center of the city's social life; its stands were capable of holding 10,000 spectators.
The emperors would place columns in the middle of the Hippodrome to adorn it, and three still remain (along with two bases, which are in the Archaeology Museum nearby). The one you can see clearly in the photos is the Obelisk of Thutmose III, which was taken from Luxor in the fourth century by Emperor Theodosius I. The emperor is depicted on the base, handing out a laurel wreath.