  Charged lepton flavour violating transitions would be a clear signal of new
physics beyond the Standard Model. Their search has been carried out in a
variety of channels, the most sensitive being those involving a muon: however
no positive evidence has been found so far. The MEG experiment has recently set
the best limit on such processes by investigating the existence of the $\mu \to
e\gamma$ decay. In the next decade several experiments are planned to pursue
the search for $\mu \to e\gamma$, $\mu \to eee$, $\mu \to e$ conversion in
nuclei, as well as on processes involving the $\tau$, to an unprecedented level
of precision. In this review we want to give a pedagogical introduction on the
theoretical motivations for such searches as well as on the experimental
aspects upon which they are based.
