The Standard Model (SM) is one of the most tested and verified physical theories of all time, present experimental observations are consistent with SM expectations. On the other hand SM can not explain many physical observations: the cosmological observations possibly infer the presence of dark matter which is clearly beyond the SM expectations; the SM Higgs model, while explaining the generation of fermion masses, can not explain the hierarchy problem and a non natural fine tuning of SM is needed to cancel out quadratic divergences in the Higgs boson mass. New physics (NP) beyond SM should hence be investigated: rising the energy above NP processes thresholds, and detecting new particles or new effects not predicted by the standard model directly, is one of the possible approaches; another approach is to make precision measurements of well known processes or looking for rare processes which involve higher order contribution from NP processes, this approach need higher luminosities with respect to the previous approach but lower beam energies. Search for Lepton Flavor Violation (LFV) in charged lepton decays is promising: neutrino physics provides indeed a clear and unambiguous evidence of LFV in the neutral lepton sector via mixing processes, which have been observed for the first time by the Homestake collaboration. We expect LFV in the charged sector as well, both in [mu] and [tau] sector, but current experimental searches for LFV processes did not find any evidence for those processes, and more results are expected to come from new experiments in the coming years.