Not yet reviewed by Pith; the record is open.

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.