Muon plays an increasingly important role in particle, nuclear, and atomic physics, and in applied research. The muon with the muon neutrino and strange and charm quarks create second generation of the Standard Model particles. Unique properties of muons, including its electric charge, mass, and lack of interaction via strong force made this particle a unique tool for discoveries of new elementary particles, including the Higgs boson, over last half a century. The prompt (by cascade transitions) and delayed (by weak muon capture) fission of heavy nuclei in muonic atoms became an important aspect of research. Use of muons as a probe particle to study various solid state samples recently developed in a separate branch of science. Muons can be used in the cold fusion for efficient energy production in the future. The studies of the processes beyond the Standard Model, the proton radius puzzle, the rare decays of the muon and its conversion into an electron and muonium into antimuonium, and hints of a difference in the anomalous magnetic moment of the muon from predicted by the Standard Model, have become hot research topics. Muons are proposed to be used in accelerators providing ultra high intensity neutrino beams which will be used for studies of neutrinos, including their oscillations, which could shed a light on matter-antimatter universe asymmetry as well as for "Higgs factories" where a large number of Higgs bosons can be produced for in depth understanding of this recently discovered particle. This book describes various aspects of modern physics involving muons.