Rotary positive displacement machines with trochoidal geometry that comprise a helical rotor that undergoes planetary motion within a helical stator are described. The rotor can have a hypotrochoidal cross-section, with the corresponding stator cavity profile being the outer envelope of the rotor as it undergoes planetary motion, or the stator cavity can have an epitrochoidal cross-section with the corresponding rotor profile being the inner envelope of the trochoid as it undergoes planetary motion. In some multi-stage embodiments, the rotor-stator geometry remains substantially constant along the axis of the rotary machine. In other multi-stage embodiments, the rotor-stator geometry varies along the axis of the rotary machine.