  Neutrino masses provide evidence of lepton flavour violation, but no
violation in the interactions among the charged leptons has been observed yet.
Many models of Physics Beyond the Standard Model (BSM) predict Charged Lepton
Flavour Violation (CLFV) in a wide spectrum of processes with rates in reach of
upcoming experiments. The experimental searches that provide the current best
limits on the CLFV searches are reviewed, with a particular emphasis on the
muon-based experiments that give the most stringent constraints on BSM
parameter space. The next generation of muon-based experiments (MEG-II, Mu2e,
COMET, Mu3e) aim to reach improvements by many orders of magnitude w.r.t. the
current best limits, thanks to several technological advancements. We review
popular heavy BSM theories, and we present the calculations of the predicted
CLFV branching ratios, focusing on the more sensitive $\mu\to e$ sector.
