Protecting Domestic Violence and Stalking Victims Act of 2013 - Amends the Brady Handgun Violence Prevention Act to revise the definition of: (1) "intimate partner" to include a dating partner and any other person similarly situated to a spouse who is protected by the domestic or family violence laws of the state or tribal jurisdiction in which the injury occurred or where the victim resides; and (2) "misdemeanor crime of domestic violence” to include the use or attempted use of physical force or a deadly weapon by a current or former intimate partner.

Prohibits: (1) the sale or other disposition of a firearm or ammunition to any person knowing or having reasonable cause to believe that such person has been convicted of a misdemeanor crime of stalking; and (2) the shipment or possession of a firearm or ammunition in interstate or foreign commerce, or the receipt of a firearm or ammunition that has been transported in interstate or foreign commerce, by an individual who has been convicted of a misdemeanor crime of stalking.