The greatest circus in Rome was the Circus Maximus, which seated around 250,000 people; another 250,000 could be accommodated on the surrounding hills, giving it a seeing capacity of 500,000, which gives you a sense for the popularity of the ludi circenses. According to tradition it was laid out by the Etruscan King Tarquinius Priscus (ruled from 614 BCE); it was certainly the oldest location for spectacles in Rome. Unfortunately, its location in the Murcian Valley, between the Palatine and Aventine hills its proximity to the River Tiber meant it had a tendency to flood (sometimes right in the middle of races), until the Romans finally built permanent and stone starting gates. It also burnt down a number of times, along with other parts of Rome.
Digital reconstructions of the Circus Maximus can be found here at Rome Reborn and here at the Circus Maximus project. This video gives a flyover of a digitally reconstructed Circus. The Circus also contained a number of temples, altars and religious shrines, as well as being a location for chariot and horse races. The arcades around it were filled with shops and were a prime location for prostitutes, fortune-tellers, and other people who had goods or themselves to sell. When the Emperor Elagabalus decided to gather up the prostitutes of Rome to give them a speech (while he himself was dressed as a woman) he apparently got them “from the Circus, the theatre, the Stadium and all other places of amusement, and from the public baths” (Historia Augusta, Elagabalus 26.3).
There were also a number of other circuses in and around Rome:
1. Circus Flaminius, from 220 BCE, which contained the temple for Pietas, as this one got built up quickly it is not clear if it was used for chariot racing and may have only been used for a set of games celebrated every 5 years, the Ludi Tauri; these involved horse racing, rather than chariot racing. Assemblies were also held here.
2. Circus Vaticanus (St. Peter’s is built over part of this). It was built by Caligula and used as his private race track. Claudius and Nero made it public, though private races were also held there. (It is also known as the Circus of Gaius and Nero.)
3. Circus near the Sessorian Palace. c. 220-230 CE; built by the Severans who were huge chariot racing fans.. Around 565 meters long (slightly shorter than the CM); about 115-125 meters wide ): width may be to accommodate larger unusual teams like elephants and camels. This was heavily modelled on the CM (like the Vatican Circus) complete with an obelisk in the centre. It may have seated c. 10,000. In 270 a new set of walls built around Rome by the emperor Aurelius cut through this circus, reducing it to c.450 metres
Many cities had circuses; the one in Jerash has been used for re-enactments; the one in Philipoppolis, modern Bulgaria is not as well preserved but still gives you a sense for its scale and setup; here are the locations of all known circuses in the Roman empire. The classic work on Roman circuses around the empire is Roman Circuses by John Humphrey (1986), an encyclopedic guide to circuses around the empire.