A multi-stage positive displacement vacuum pump and a method of differentially pumping multiple vacuum chambers using such a pump is disclosed. The pump comprises a housing for housing at least one rotor mounted for rotation on a corresponding at least one shaft and for pumping gas through the multiple stages for output through an exhaust. The housing comprises a first inlet configured to admit gas to an inlet stage of said vacuum pump, and a further inlet configured to admit gas to an intermediate stage of said vacuum pump. The pump is configured such that gas admitted through each of the first and further inlets is pumped together as a combined gas flow by at least one stage of said vacuum pump downstream of said intermediate stage.