  Lepton flavor violation (LFV) has been discovered in the neutrino sector by
neutrino oscillation experiments. The minimally extended Standard Model (SM) to
include neutrino masses allows LFV in the charged sector (CLFV) at the
loop-level, but at rates that are too small to be experimentally observed.
Lepton number violation (LNV) is explicitly forbidden even in the minimally
extended SM, where the observation of a LNV process would be unambiguous
evidence of physics beyond the SM. The search for the LNV and CLFV process
$\mu^- + N(A,Z) \to e^+ + N'(A,Z-2)$ (referred to as $\mu^- \to e^+$
conversion) is a complementary LNV channel to neutrinoless double beta decay
searches, sensitive to potential flavor effects in the neutrino mass generation
mechanism. A theoretical motivation for $\mu^- \to e^+$ conversion is presented
along with a review of the status of past $\mu^- \to e^+$ conversion
experiments and future prospects. Special attention is taken to understanding
an uncertain and potentially dominant background for these searches, radiative
muon capture, and its potential impacts on current and future $\mu^- \to e^+$
searches.
