A number of experiments with muons are limited by the poor phase space quality of the muon beams currently available. The muCool project aims at developing a phase-space cooling method to transform a surface \mu^+μ+ beam with 4 MeV energy and 1 cm size into a slow muon beam with eV energy and 1 mm size. In this process the phase space is reduced by a factor of 10^{9}-10^{10}109−1010 with efficiencies of 2\cdot 10^{-5}-2\cdot 10^{-4}2⋅10−5−2⋅10−4. The beam is then re-accelerated to keV-MeV energies. Such a beam opens up new avenues for research in fundamental particle physics with muons and muonium atoms as well as in the field of \muμSR spectroscopy.