A pump unit for the metered delivery of fuel to internal combustion engines. The pump unit comprises a housing which comprises a longitudinal bore. Located in the longitudinal bore is an overflow valve, via which fuel flows back through a channel into a fuel tank. The passage can be opened or closed by a spring-loaded closing element. Fastened to the valve shaft of the overflow valve is a ring fitting. In the longitudinal bore of the housing there is an additional thread section, via which air flows out through vent gaps into a cavity of the ring fitting.