As a result of advances in particle accelerator technology and difficulties in building new nuclear reactors, increasingly ambitious applications of particle accelerator-driven spallation targets have been proposed in recent years. The simplest applications are the spallation neutron sources needed for basic nuclear sciences, with proton beams in the 1- to 5-MW range to be driven into targets of lead, mercury, or tungsten to produce neutron fluxes higher than is practical with nuclear reactors. On a much larger scale, the proposed accelerator production of tritium would use a 170-MW proton beam to generate sufficient neutrons to produce ~3 kg tritium/yr, based on neutron capture in a 3He feedstock. Other proposals include the use of subcritical neutron multiplication, using waste actinides and/or fertile actinides to transmute nuclear wastes or support alternate fuel cycles. The basic technology and technical aspects of the numerous-proposed applications are described. Fundamental relationships regarding machine efficiencies, neutron production, and subcritical multiplication are provided and utilized to cross-compare concepts.