A muon collider would be both a precision and a powerful discovery machine, because it can offer collisions of point-like particles at very high energies, strongly exceeding the energy reach of other lepton colliders. It can even match the discovery of a proton collider with much higher energy, since the muon collision energy is fully available at constituent level unlike for the protons. However, the need for high luminosity faces technical challenges which arise from the short muon lifetime at rest and the difficulty of producing large numbers of muons in bunches with small emittance. Addressing these challenges requires the development of innovative concepts and demanding technologies. In view of the potential importance of muon colliders for the future of high-energy physics it is timely to start an R&D programme now.