In muon physics at low energies (keV-MeV), spectroscopy of muonium atoms takes a significant role in fundamental physics at precision and intensity frontiers. Efforts are invested nowadays in testing theoretical calculations of the Standard Model with a precision down to $\approx1$ ppb. New measurements for two energy intervals of the muonium, the 1s hyperfine structure and the 1s-2s interval, are designed utilizing new generations of muon sources and new experimental techniques for microwave and laser spectroscopy. These ongoing experiments will also determine the muon mass and magnetic moment with unprecedented precisions. This paper first reviews the current experimental activities on precision spectroscopy of the muonium atoms. The latter part focuses on new measurements of the 1s hyperfine splitting by the MuSEUM collaboration at J-PARC. Future measurement plans using a rectangular shaped microwave cavity are also described.