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Fast radio bursts (FRBs), millisecond-duration bursts prevailing in the radio sky, are the latest big puzzle in the universe and have been a subject of intense observational and theoretical investigations in recent years. The rapid accumulation of the observational data has painted the following sketch about the physical origin of FRBs: They predominantly originate from cosmological distances so that their sources produce the most extreme coherent radio emission in the universe; at least some, probably most, FRBs are repeating sources that do not invoke cataclysmic events; and at least some FRBs are produced by magnetars, neutron stars with the strongest magnetic fields in the universe. Many open questions regarding the physical origin(s) and mechanism(s) of FRBs remain. This article reviews the phenomenology and possible underlying physics of FRBs. Topics include: a summary of the observational data, basic plasma physics, general constraints on FRB models from the data, radiation mechanisms, source and environment models, propagation effects, as well as FRBs as cosmological probes. Current pressing problems and future prospects are also discussed.