A multi-stage vacuum pump comprises a first piston in a first cylinder defining a first stage pumping chamber and a second piston in a second cylinder defining a second stage pumping chamber. The pump has a spool valve between the first and second cylinders for controlling flow from the first stage pumping chamber to the second stage pumping chamber. Either the first cylinder and the first piston or the second cylinder and the second piston being stepped so as to define a third stage pumping chamber. The spool valve is arranged to control the flow from the second stage pumping chamber to the third stage pumping chamber.