In the past decade, CubeSats have revolutionized small spacecraft missions. These miniature satellites began as educational projects but have lowered the bar for access to space and enabled research institutions and companies to launch technology demonstration and science missions in low Earth orbit. Propulsion systems small enough to fit in a CubeSat can extend the benefits of CubeSats beyond low Earth orbit, and potentially even allow for small-scale interplanetary missions. Propulsion systems designed for CubeSats must overcome severe restrictions in their chemistry, dimensions, mass and operation scheme for the sake of fitting within the CubeSat deployer and conforming to CubeSat specifications. This research presents a novel concept for small satellite propulsion based on the electrolysis of water. These systems are designed to ensure the safety of the launch vehicle and overcome the restrictions imposed by operating as a secondary payload by avoiding the use of hazardous materials, pressure vessels and explosives. Numerical analyses are used to predict the performance of the propulsion system. Vacuum chamber experiments on a prototype of the propulsion system are conducted to determine the performance of the system. An analysis of the attitude dynamics and operation of a satellite with an electrolysis propulsion system are presented. The propulsion system as well as the attitude control of the spacecraft are aided by the spacecraft's spin about its major axis of inertia. Energy damped by the water carried on board keeps the satellite stable and damps nutation caused by external torques and the use of the propulsion system. Several applications are presented for low earth orbit as well as interplanetary CubeSats. The design of a mission to navigate a CubeSat to lunar orbit as part of NASA's CubeQuest Challenge is detailed. Prospects for broader applications of this work involving distributed exploration using in-situ water are identified.