The present invention provides a multistage vacuum pump comprising at least a low pressure first stage and at least a high pressure second stage. At least one of the stages has at least one suction inlet for admitting gas to be pumped, and at least one of the stages has at least one delivery outlet that is open to the outside for exhausting pumped gas. The first and second stages communicate with each other in order to pass gas from the first stage to the second stage. The first stage operates at a flow rate that is smaller than that of the second stage. For an oil-seal rotary vane positive-displacement pump, the oil is injected into the stage having the greater flow rate.