Muon number violating processes in nuclei

The flavour violating neutrinoless muon decays in the presence of nuclei, are discussed. We focus on the theoretical aspects of $\mu^-(A,Z)\to e^-(A,Z)^*$ (muon-to-electron conversion), one of the most prominent flavour changing reactions, emphasizing its connection with the physics beyond the standard model. This process offers the most severe limits of the lepton flavour violation. By using the nuclear transition matrix elements calculated with several methods, and the recent experimental data of the branching ratio $R_{\mu e^-}$, we determine limits for the flavour changing parameters entering the elementary sector part of $R_{\mu e^-}$. These results are discussed in view of the ongoing experiment at PSI and the designed at Brookhaven, which are expected to push down by some orders of magnitude the experimental sensitivity the next few years with the hope to see ``new physics''.