  Charged-lepton flavour-violating (cLFV) processes offer deep probes for new
physics with discovery sensitivity to a broad array of new physics models -
SUSY, Higgs Doublets, Extra Dimensions, and, particularly, models explaining
the neutrino mass hierarchy and the matter-antimatter asymmetry of the universe
via leptogenesis. The most sensitive probes of cLFV utilize high-intensity muon
beams to search for $\mu \rightarrow e$ transitions. We summarize the status of
muon-cLFV experiments currently under construction at PSI, Fermilab, and
J-PARC. These experiments offer sensitivity to effective new physics mass
scales approaching O($10^4$) TeV/c$^2$. Further improvements are possible and
next-generation experiments, using upgraded accelerator facilities at PSI,
Fermilab, and J-PARC, could begin data taking within the next decade. In the
case of discoveries at the LHC, they could distinguish among alternative
models; even in the absence of direct discoveries, they could establish new
physics. These experiments both complement and extend the searches at the LHC.
