Some theoretical aspects of a nonzero value for the neutrino rest mass and its possible implications for physics are discussed. The nature of the neutrino mass is analyzed, as well as the physical consequences that may derive from the existence of new helicity states for the neutrino or from lepton charge nonconservation if the mass is of Dirac or Majorana character, respectively. Massive neutrinos are examined in the context of grand unified theories combining the weak, strong, and electromagnetic interactions. Searches for neutrino-mass effects in $\beta$ decay and for neutrino oscillations are reviewed. Several astrophysical effects of the neutrino mass are described: solar-neutrino oscillations, the decay of primordial neutrinos, the feasibility of detecting massive primordial neutrinos experimentally. The predictions of big bang theory regarding the neutrino number density in the universe are analyzed, and a discussion is given of the influence neutrino oscillations might have on the neutrino density and on cosmological nucleosynthesis.