A multi-degree-of-freedom electromagnetic machine that may be operated as a motor, a generator, or a motor-generator, includes a first structure and a second structure. The first structure comprises a first conductor, a second conductor, and a third conductor, each of which follows a different trajectory. The first, second, and third conductors together form a general shape of a surface. The second structure is disposed adjacent to the first structure and includes a magnet that emanates a magnetic field. The magnet has at least one of its magnetic poles facing the surface. A Lorentz force affects relative movement between the first structure and the second structure when the magnetic field that emanates from the at least one magnetic pole interacts with electrical currents within any of the electrical conductors.