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Circus of Maxentius is the best preserved circus from time of ancient Rome. It was designed and built with great care but it didn't see much use. Because of that we can today see, in great detail, how ancient circuses looked like.
Contemporary circus is relatively new subtype of circus (some 50 years relative to 250 of a traditional circus) which mixes elements of its predecessor with higher art and other type of media.
Although almost nothing has left of Circus Flaminius today, we know that it existed and that it was important public place for hundreds of years where games and other public businesses were held.
Circus of Nero began as a place for public trainings of Caligula and Nero and became a place of martyrdom of many early Christians. When it was abandoned it became a place of not one but two basilicas.