2Abstract. Muon Colliders have unique technical and physics advantages and disadvantages when compared with both hadron and electron machines. They should thus be regarded as complementary. Parameters are given of 4 TeV and 0.5 TeV high luminosity µ+µ−colliders, and of a 0.5 TeV lower luminosity demonstration machine. We discuss the various systems in such muon colliders, starting from the proton accelerator needed to generate the muons and proceed