A longstanding question in physics is whether antimatter falls up or down. The equivalence principle of general relativity tells us it falls down; however, if it falls up, a simpler cosmology, with no inflation or baryon asymmetry, and no need for dark matter and energy, may be possible. Alternatively, it may fall down at nearly the same rate as matter, violating the equivalence principle only slightly, and pointing towards a future quantum theory of gravity. For these reasons, several experiments at CERN seek to manufacture antihydrogen and measure its gravitational acceleration. We explore the possibility of similar experiments on muonium, a hydrogen-like atom whose nucleus is an antimuon rather than a proton. The short muon lifetime imposes stringent limits on the required interferometer technology. Meeting them may be feasible using nanotechnology and modern laser feedback techniques. The measurement may thus be feasible using muonium sources at J-PARC, RAL, TRIUMF, or PSI. If so, the outcome could point to a new and better cosmology and theory of gravity. Or it could confirm the equivalence principle in a new realm, at long last laying the question to rest.