Accelerated muon beams have been considered for the next-generation studies of high-energy lepton–antilepton collisions and neutrino oscillations. However, high-brightness muon beams have not yet been produced. The main challenge for muon acceleration and storage stems from the large phase-space volume occupied by the beam, derived from the production mechanism of muons through the decay of pions. The phase-space volume of the muon beam can be decreased through ionization cooling. Here we show that ionization cooling leads to a reduction in the transverse emittance of muon beams that traverse lithium hydride or liquid hydrogen absorbers in the Muon Ionization Cooling Experiment. Our results represent a substantial advance towards the realization of muon-based facilities that could operate at the energy and intensity frontiers.