  Searches for charged lepton flavor violation (CLFV) offer deep probes for a
range of new physics scenarios, such as super-symmetric models, theories
involving scalar leptoquarks or additional Higgs doublets, and models
explaining the neutrino mass hierarchy and the matter-antimatter asymmetry of
the universe via leptogenesis. The coherent, neutrinoless conversion of a muon
to an electron in the field of a nucleus, $\mu^{-}+N\rightarrow e^{-} +N$, is
one example of a muonic CLFV process which has sensitivity to this new physics.
This paper details three experiments: COMET, DeeMe and Mu2e which will search
for $\mu^{-}+N\rightarrow e^{-} +N$ in the coming decade. These experiments
offer sensitivity up to an effective new physics mass scale of
$\mathcal{O}(10^{4}$ TeV/c$^{2}$), going far beyond what can be achieved in
direct, collider-based, searches. The theoretical motivation, designs and
anticipated timelines for these three experiments are presented. These
experiments are a crucial part of a global search for CLFV. Continued support
for all planned experimental searches for muonic CLFV is strongly encouraged.
