The aim of the invention is to provide a rotary displacement machine, whereby turbulence in the supply area, i.e. the area where the medium to be displaced passes from the supply channels into the displacement chambers, is prevented to a large extend and the filling of the displacement chambers is improved. Disclosed is a rotary displacement machine with at least two displacement gearwheels with external teeth for displacing a fluid and with at least one supply channel with at least one supply mouth in the area of the displacement gearwheels for supplying the fluid to the displacement gearwheels. According to the invention, at least two supply mouths are provided and each supply mouth is embodied in such a way that fluid that exits the supply mouth when the rotary displacement machine is operated in a correct manner can either only reach the teeth of only one displacement gearwheel or the teeth of such displacement gearwheels which rotate in the same direction.