A rotary apparatus includes a stationary stator having an internal cylindrical chamber. A rotor is rotatably mounted in the chamber. There are cam-like annular surfaces on the stator to each side of the rotor. A plurality of vane assemblies are slidably mounted on the rotor. The vane assemblies each have two outer ends. These slidingly engage the cam-like annular surfaces, whereby gases between the vane assemblies are compressed and expand between the cam-like surfaces and the rotor as the rotor and vanes rotate relative to the stator. Preferably each of the vane assemblies has two separate components, each outer end being on one of the components. The apparatus has a conduit for conducting compressed gases between the two components of each of the vane assemblies and thereby biasing each of the components towards one of the cam-like annular surfaces. Preferably there is an intake system for intaking gases between the vane assemblies and between the cam-like surfaces and the rotor and an exhaust systems for conducting exhaust gases away from the spaces. Each of the intake and exhaust systems preferably includes a reed valve. There may be a rotatable annular housing within the stator between the stator and the rotor. The components of the vane assemblies engage the housing, the housing rotating with the rotor and the vane assemblies.