Handgun Licensing and Registration Act of 2013 - Amends the federal criminal code to direct the Attorney General to establish a federal system for the licensing and registration of all handguns in the United States, including a method for easily retrieving information sufficient to identify each resident of a state who owns, possesses, or controls a handgun and each such handgun.

Prohibits the ownership, possession, or control of a handgun in a state by a person who: (1) is not licensed under such system; and (2) has not registered the handgun with a federal, state, or local law enforcement agency. Provides for a fine and/or imprisonment not less than 15 years for violations. Prohibits the court from suspending a sentence of imprisonment or imposing a probationary sentence for anyone who knowingly violates such prohibition.

Exempts states the Attorney General certifies have in effect such a licensing and registration system that imposes criminal penalties on anyone who owns, possesses, or controls a handgun and who: (1) has not completed training in firearms safety; (2) is not licensed by the state to possess a handgun; or (3) has not registered the handgun with a federal, state, or local law enforcement agency.