A compact fusion reactor is one that has a higher power density and, for the same total power, is significantly smaller than a conventional magnetic fusion reactor. This survey reviews the principal physics and technology employed by compact fusion power plants. Each of these concepts has been proposed as a fusion power source and rudimentary power plant designs exist. The concepts reviewed are: compact reversed-field pinch reactors, the Ohmically Heated Toroidal Experiment reactor, TRACT, field-reversed mirror reactor, spheromak, field-reversed theta pinch, compact tokamak reactors, dense Z-pinch reactor, imploding liner reactors, and the wall-confined fusion reactor.