Abstract: An air gun includes an air pump with a pumping piston mounted to reciprocate within a pump cylinder, which is mounted on the barrel of the gun to pivot about an axis transverse to the longitudinal axis of the barrel so that as the pump cylinder is moved back and forth around the pivot, the cylinder and piston reciprocate relative to each other to pump air into a high pressure housing carried by the pump cylinder. A discharge conduit is also provided for releasably connecting the high pressure housing to the breach end of a gun barrel when the pump is moved toward the barrel. A firing valve in the discharge conduit releases air from the high pressure housing into the breach end of the barrel. Preferably, a floating differential piston disposed to reciprocate in a high pressure housing divides the housing into a storage chamber and a high pressure chamber.