Historically, progress in particle physics has largely been determined by development of more capable particle accelerators. This trend continues today with the recent advent of high-luminosity electron-positron colliders at KEK and SLAC operating as "B factories," the imminent commissioning of the Large Hadron Collider at CERN, and the worldwide development effort toward the International Linear Collider. Looking to the future, one of the most promising approaches is the development of muon-beam accelerators. Such machines have very high scientific potential, and would substantially advance the state-of-the-art in accelerator design. A 20-50 GeV muon storage ring could serve as a copious source of well-characterized electron neutrinos or antineutrinos (a Neutrino Factory), providing beams aimed at detectors located 3000-7500 km from the ring. Such long baseline experiments are expected to be able to observe and characterize the phenomenon of charge-conjugation-parity (CP) violation in the lepton sector, and thus provide an answer to one of the most fundamental questions in science, namely, why the matter-dominated universe in which we reside exists at all. By accelerating muons to even higher energies of several TeV, we can envision a Muon Collider. In contrast with composite particles like protons, muons are point particles. This means that the full collision energy is available to create new particles. A Muon Collider has roughly ten times the energy reach of a proton collider at the same collision energy, and has a much smaller footprint. Indeed, an energy frontier Muon Collider could fit on the site of an existing laboratory, such as Fermilab or BNL. The challenges of muon-beam accelerators are related to the facts that i) muons are produced as a tertiary beam, with very large 6D phase space, and ii) muons are unstable, with a lifetime at rest of only 2 microseconds. How these challenges are accommodated in the accelerator design will be described. Both a Neutrino Factory and a Muon Collider require large numbers of challenging superconducting magnets, including large aperture solenoids, closely spaced solenoids with opposing fields, shielded solenoids, very high field (~;40-50 T) solenoids, and storage ring magnets with a room-temperature midplane section. Uses for the various magnets will be outlined, along with R&D plans to develop these and other required components of such machines.