Lepton Flavour Violation in the charged lepton sector (CLFV) is forbidden in the Minimal Standard model and strongly suppressed in extensions of the model to include finite neutrino mixing. On the other hand, a wide class of Supersymmetric theories, even coupled with Grand Unification models (SUSY-GUT models), predict CLFV processes at a rate within the reach of new experimental searches operated with high resolution detectors at high intensity accelerators. As the Standard model background is negligible, the observation of one or more CLFV events would provide incontrovertible evidence for physics beyond Standard model, while a null effect would severely constrain the set of theory parameters. Therefore, a big experimental effort is currently (and will be for incoming years) accomplished to achieve unprecedented sensitivity on several CLFV processes. In this paper we review past and recent results in this research field, with focus on CLFV channels involving muons and tau's. We present currently operating experiments as well as future projects, with emphasis laid on how sensitivity enhancements are accompanied by improvements on detection techniques. Limitations due to systematic effects are also discussed in detail together with the solutions being adopted to overcome them.