  We show that lepton number violating muon decays, \mu^+ -> e^+ + \nu_e-bar +
\nu_i-bar (i=e, \mu or \tau), can consistently explain the neutrino anomaly
reported by the LSND experiment. Two effective operators in the Standard Model
are identified which lead to just such decays and no other processes. The scale
of new physics \Lambda must be relatively low, \Lambda < 500 GeV. Extensions of
the Standard Model which realize these effective operators are presented. Since
new physics affects only the decay of the muon, and not of \pi^+-, our scenario
predicts a null result for \nu_\mu-bar - \nu_e-bar oscillation searches at the
Fermilab mini--BOONE experiment. Models which realize these effective
operators, while consistent with all available data, can be tested in the near
future through (i) discovery of new scalar particles with masses below about
500 GeV, (ii) small but observable deviations in e^+ e^- -> \mu^+ \mu^- and e^+
e^- -> \nu \nu \gamma cross sections, (iii) observable corrections to the muon
g-2, and (iv) lepton number violating Z^0 decays with branching ratios of order
10^{-7}.
