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Muonium, a hydrogen-like atom consisting of a positive muon and an electron, plays a unique role in testing the Standard Model and searching for new physics. Spectroscopy of muonium provides a rigorous validation of quantum electrodynamics through a comparison with theoretical predictions. At J-PARC MLF MUSE, a series of microwave spectroscopy experiments of the ground-state hyperfine splitting in muonium is in progress. On the other hand, the application of muonium as a matter wave is limited because the surface muon and its slowing down in a degrader can not maintain sufficient coherence. Low-energy muons obtained at MUSE through the laser ionization of muonium can be employed to use slow muonium with minimal temporal and spatial spread. A muonium interferometer possesses various potential applications, such as a measurement of the Berry phase and a precise measurement of the muon mass. In this talk, we discuss muonium spectroscopy and interferometry.