A fuel injection pump for internal combustion engines has a single rotating and reciprocating piston which alternately aspirates fuel from a fuel storage compartment and delivers it under pressure to one of several injection lines. Fuel aspiration takes place through a plurality of fuel channels which terminate in the wall of the pump cylinder and whose termini are opened by appropriate grooves in the piston surface. The individual fuel channels are in mutual communication via one or more connection channels and are also connected to the fuel storage compartment via a suction conduit.