Space infrastructure such as the International Space Station and proposed Lunar Gateway motivate technologies for on-orbit servicing in microgravity. A key challenge for routine inspection of infrastructure is the mobility of extravehicular free flyers relative to client spacecraft. Propellant-free actuators allow free flyers to use sustainable power sources such as batteries and solar panels. These power sources eliminate launch mass for consumable propellant and, therefore, support routine inspection and tasks with indefinite durations. Electromagnetic technologies provide propellant-free actuation relative to conductive, non-ferromagnetic clients. Aluminum Micrometeoroid and Orbital Debris shielding, which is non-ferromagnetic, provides the required conductive surface for many resident and proposed spacecraft. This dissertation presents approaches for multi-degree-of-freedom relative positioning as well as analysis of component technologies for electromagnetic actuation. First, an actuator array combines eddy-current and electrostatic actuators to provide three-degree-of-freedom translation relative to a client surface. Second, the addition of eddy-current actuators forms an array for six-degree-of-freedom translation and rotation relative to a client surface. Analytical, numerical, and finite element models of individual actuators inform simulation of actuator allocation and position control of a free flyer. Third, this work demonstrates open- and closed-loop operation of rotating eddy-current actuators. Fourth, this work experimentally validates a model for eddy-current repulsion and attraction using a translating permanent magnet. The model informs a design study for use of this actuation approach for relative repulsion. These contributions support the utility of propellant-free electromagnetic technologies for mobility of microgravity free-flyers at millimeter and centimeter separations from conductive, non-ferromagnetic clients. This work provides a foundation for future applications (e.g., in-space servicing, assembly, and manufacturing) that combine eddy-current and/or electrostatic actuators for multi-DOF positioning relative to conductive surfaces.