A significant worldwide R & D effort is presently directed toward solving the technical challenges of producing, cooling, accelerating, storing, and eventually colliding beams of muons. Its primary thrust is toward issues critical to a Neutrino Factory, for which R & D efforts are under way in the U.S., via the Neutrino Factory and Muon Collider Collaboration (MC); in Europe, centered at CERN; and in Japan, at KEK. Under study and experimental development are production targets handling intense proton beams (1-4 MW), phase rotation systems to reduce beam energy spread, cooling channels to reduce transverse beam emittance for the acceleration system, and storage rings where muon decays in a long straight section provide a neutrino beam for a long-baseline (3000 km) experiment. Critical experimental activities include development of very high gradient normal conducting RF (NCRF) and superconducting RF (SCRF) cavities, high-power liquid-hydrogen absorbers, and high-field superconducting solenoids. Components and instrumentation that tolerate the intense decay products of the muon beam are being developed for testing. For a high-luminosity collider, muons must be cooled longitudinally as well as transversely, requiring an emittance exchange scheme. In addition to the experimental R & D effort, sophisticated theoretical and simulation tools are needed for the design. Here, the goals, present status, and future R & D plans in these areas will be described.