Advances in ionization cooling, phase space manipulations, and technologies to achieve high brightness muon beams are stimulating designs of high-luminosity energy-frontier muon colliders. Simulations of Helical Cooling Channels (HCC) show impressive emittance reductions, new ideas on reverse emittance exchange and muon bunch coalescing are being developed, and high-field superconductors show great promise to improve the effectiveness of ionization cooling. Experiments to study RF cavities pressurized with hydrogen gas in strong magnetic fields have had encouraging results. A 6-dimensional HCC demonstration experiment is being designed and a 1.5 TeV muon collider is being studied at Fermilab. Two new synergies are that very cool muon beams can be accelerated in ILC RF structures and that this capability can be used both for muon colliders and for neutrino factories. These advances are discussed in the context of muon colliders with small transverse emittances and with fewer muons to ease requirements on site boundary radiation, detector backgrounds, and muon production.
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