Here is a list of some popular first and third person games where you play the role of a criminal, or mobster, or law enforcement searching for criminals. Games of this genre such as Grand Theft Auto 3 and Hitman 2: Silent Assassin have exploded onto video games over the last couple years.
Grand Theft Auto 3 is a third person action game where you play the role of a common criminal who's looking to move up in the criminal world and take revenge all to increase your stature in the criminal world
Hitman 2: Silent Assassin marks the return of Codename 47 a hitman brought out of retirement after a brutal murder. Storyline and game play are excellent and use of stealth over guns blazing is needed at the right times.
In Mafia the player assumes the role of a gangster in an American city of the 1930s, and gets a chance to live through gangster's rise and fall. In a spell-binding story of human hunger for power, he will experience everything that was taking place in the underworld during this stormy period: high-speed car chases, bootlegging, assassinations, and bank robberies. - Illusion Softworks
The follow up to the smash Grand Theft Auto 3, Vice City takes place in the 1980s in a city somewhat similiar to Miami, called of course, Vice City.
You are Max Payne, a fugitive undercover cop framed for murder and hunted by the police.
With the real actors lending their voices to the game, fans of the show should not be disappointed, except perhaps by the short number of missions.
Hitman Codename 47 is a third-person action-adventure game that combines stealth, strategy, and shooting. Play the role of a mysterious assassin who has no idenity
Post Mortem is set during the 1920s in Paris and you play the role of an American private Detective hired to solve the mystery. More of a first person adventure game rather than action.
Kingpin Life of Crime is a gang simulation set in an urban environment. Built using the Quake 2 game engine, it contains over 20 levels.