Abstract: 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\rightarrow e\gamma$ decay. In the next decade several experiments are planned to pursue the search for $ \mu\rightarrow e\gamma$, $ \mu\rightarrow eee$, $\mu \rightarrow e$ 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.