Rotary positive displacement machines based on trochoidal geometry that includes a helical rotor that undergoes planetary motion relative to a helical stator are described. The rotor can have a hypotrochoidal-based cross-sectional shape, with the corresponding stator cavity cross-sectional shape being the outer envelope of the rotor cross-sectional shape as it undergoes planetary motion, or the stator cavity can have an epitrochoidal-based cross-sectional shape with the corresponding rotor cross-sectional shape being the inner envelope of the stator cross-sectional shape as it undergoes planetary motion. Such machines can be configured so that the stator axis is spaced from the rotor axis, the rotor is configured to spin about its axis and the stator is configured to spin about its axis, and/or the rotor and the stator are held at a fixed eccentricity so that the rotor undergoes planetary motion relative to the stator, but does not orbit.