Liquid-propellant rocket engines are capable of on-command variable thrust or thrust modulation, an operability advantage that has been studied intermittently since the late 1930s. Throttleable liquid-propellant rocket engines can be used for planetary entry and descent, space rendezvous, orbital maneuvering including orientation and stabilization in space, and hovering and hazard avoidance during planetary landing. Other applications have included control of aircraft rocket engines, limiting of vehicle acceleration or velocity using retrograde rockets, and ballistic missile defense trajectory control. Throttleable liquid-propellant rocket engines can also continuously follow the most economical thrust curve in a given situation, as opposed to making discrete throttling changes over a few select operating points. The effects of variable thrust on the mechanics and dynamics of an liquid-propellant rocket engine as well as difficulties and issues surrounding the throttling process are important aspects of throttling behavior. This review provides a detailed survey of liquid-propellant rocket engine throttling centered around engines from the United States. Several liquid-propellant rocket engine throttling methods are discussed, including high-pressure-drop systems, dual-injector manifolds, gas injection, multiple chambers, pulse modulation, throat throttling, movable injector components, and hydrodynamically dissipative injectors. Concerns and issues surrounding each method are examined, and the advantages and shortcomings compared.