During the Hunger Games, the tributes are brought to the City Circle. The tributes ride their chariots through the Avenue of The Tributes in their respective district-themed costumes and then listen to a welcoming speech given by President Coriolanus Snow. This is where Katniss Everdeen first showed off her trademark 'Girl on Fire' look at the beginning of the 74th Hunger Games. In the film, hundreds of Capitol citizens are seen standing in City Circle the night before the 74th Hunger Games begin, chanting, "Hunger! Hunger!"
This is also where, during Mockingjay, a group of toddlers, children, and teenagers (including Primrose Everdeen) were killed by bombs falling on them while the Capitol was swarmed by rebels during the second rebellion.
Many citizens of the Capitol could be seen gathering in the City Circle the night before the games to join in with the festivities.