To provide a multi-stage Roots pump which prevents interference between a Roots rotor and a housing caused by thermal expansion without dropping the pump efficiency.

The multi-stage Roots type pump is structured so that the ratio of the linear expansion coefficient α2 of its rotary shaft to the linear expansion coefficient α1 of the housing meets the formula α2/α1=(T1-T0)/(T2-T0), where α1(/K) is the linear expansion coefficient of the rotary shaft, α2(/K) is the linear expansion coefficient of the housing, T0(K) is the temperature of the rotary shaft and housing when the Roots rotor is out of rotation, T1(K) is the temperature of the rotary shaft when the Roots rotor is rotating, and T2(K) is the temperature of the housing when the Roots rotor is rotating.