Rome's birthday is celebrated on April 21, commemorating the date of Rome's founding in 753BC. Rome will hold several festivals, concerts, and special events this weekend and on April 21. Historical societies put on gladiator displays and performances in and around the forum area. There's usually a spectacular fireworks display by the Tiber River.
According to legend, the twin brothers Romulus and Remus were the founders of Rome. Following the murder of Remus, Romulus became the first king of Rome. The traditional date of Romulus' sole reign and the founding of Rome is April 21, 753 BC. Read more on the Founding of Rome on about.com's Ancient History site. Eternally Cool has a good blog about the birthday of Rome and photos of a performance re-enacting the ancient story.
Related Italy Travel: Top Ancient Rome Sites | April Italian Festivals | May Italian Festivals
Photo of Rome birthday garden display © 2008 by James Martin, Europe for Visitors, taken at the city of Rome's garden on a SecretGardensItaly tour