The nuclear fusion reaction can be catalyzed in a suitable fusion fuel by muons (heavy electrons), which can temporarily form very tightly bound mu-molecules. Muons can be produced by the decay of negative pions, which, in turn, have been produced by an accelerated beam of light ions impinging on a target. Muon-catalyzed fusion is appropriately called “cold fusion” because the nuclear fusion also occurs at room temperature. For practical fusion energy generation, it appears to be necessary to have a fuel mixture of deuterium and tritium at about liquid density and at a temperature of the order of 1000 K. The current status of muon-catalyzed fusion is limited to demonstrations of scientific breakeven by showing that it is possible to sustain an energy balance between muon production (input) and catalyzed fusion (output). Conceptually, a muon-catalyzed fusion reactor is seen to be an energy amplifier that increases by fusion reactions the energy invested in nuclear pion-muon beams. The physical quantity that determines this balance is Xμ, the number of fusion reactions each muon can catalyze before it is lost. Showing the feasibility of useful power production is equivalent to showing that Xμ can exceed a sufficiently large number, which is estimated to be ∼104 if standard technology is used or ∼103 if more advanced physics and technology can be developed. Since a muon can be produced with current technology for an expenditure of ∼5000 MeV and 17.6 MeV is produced per fusion event, it follows that Xμ ≈ 250 would be a significant demonstration of scientific breakeven. Current experiments have measured Xμ 150. Therefore, the energy cost of producing muons must be reduced substantially before muon-catalyzed fusion reactors could seriously be considered. The physics of muon-catalyzed fusion is summarized and discussed. Muon catalysis is surveyed for the following systems: proton-deuteron, deuteron-deuteron, deuteron-triton, and non-hydrogen elements. The idea of muon catalysis in a plasma medium is also presented. The formation of mu-atoms and mu-molecules and their disintegration in a condensed plasma are calculated. It seems that in a homogeneous plasma, there are no values of temperature and density appropriate for achieving the desired Xμ ≈ 1000. New ideas that might lead to the goal of 1000 fusions per muon by the use of laser beams or selective electromagnetic radiation are suggested.