What is the name of the city Batman lives in?

Gotham City.

Batman, the fictional superhero appearing in American comic books published by DC Comics, lives in Gotham City. Gotham is depicted as a dark, crime-ridden locale, which Batman tirelessly protects. The character of Batman, whose real name is Bruce Wayne, was first introduced in Detective Comics #27 in 1939. His base of operations is Wayne Manor, which includes the Batcave located beneath the manor, where Batman keeps his Batmobile, Batcomputer, and other crime-fighting equipment.