Rolling turbines are fluid motors that may be used in pico hydro applications at very small hydraulic heads of below 1 m and can operate even with untapped (waste) water. The present study proposes a theory of their operation, based on the hydrodynamic stability of vortices. A solid body vortex may be transformed due to viscosity into a more stable potential vortex. Circulation generated in the vortex core is transmitted to a rotor. The rotor, which is the key element, performs a precession motion, rotating concurrently around its axis while processing around another (precession) axis. The mechanical power output is drawn from the precession motion via a shaft. The theoretical results were experimentally validated with three variants of the rotors: one truncated cone and two spherical rotors. It was concluded that the tested rolling turbines operate with an efficiency of 43% to 48% at a hydraulic head of between 0.5 and 1.6 m.