A vane machine for expansion or compression of gases, air, engine exhaust, vapors or a mixture thereof, includes a housing with a cylindrical space having inlet and outlet ports, a shaft offset in parallel or eccentric relative to a central housing axis and first and second circular discs on the shaft mutually offset in parallel. Slides are guided by the circular discs and displaceable in direction of an inner housing wall. Vane cells are formed by two neighboring slides and an adjoining region of the wall and volumes of the vane cells in vicinity of the inlet and outlet ports differ. To obtain reliability and efficiency, each circular disc has a plurality of circular arcuate slots, each slide can have a circular arc shape on an end facing the housing, and the circular arcuate part of each slide moves into circular arcuate slots of first and second circular discs.