A rotary piston machine having a housing (1) provided with a cavity (2) comprising a rotor (6', 6) rotating around a first axis and a compression plate (16) dividing said cavity into separate volumes, the machine further being provided with at least one inlet port and one outlet port. Said cavity (2) is provided with a spherical wall portion (3) and is closed by means of a top cover (13). The rotor is provided with a conical part (6) reaching into said cavity (2) which conical part is provided with a slot (15) through the virtual tip of the cone perpendicular to the base of the same. The compression plate (16) is provided with a circular edge portion having essentially the same radius as the spherical wall portion (3) of said cavity and is arranged sliding in a pivoting movement in said slot (15). The plate (16) has further a straight edge portion (22) arranged at the level of the virtual tip of the cone and is arranged for rotation in said cavity in a plane parallel to the top cover (13) which compression plate during operation is thus rotating at the same speed as the cone around said first axis while simultaneously pivoting around a second axis perpendicular to said first axis.