  The coherent and incoherent channels of the neutrinoless muon to electron
conversion in nuclei, $\mu^- (A,Z) \to e^- (A,Z)^*$, are studied throughout the
periodic table. The relevant nuclear matrix elements are computed by explicitly
constructing all possible final nuclear states in the context of the
quasi-particle RPA. The obtained results are discussed in view of the existing
at PSI and TRIUMF experimental data for $^{48}Ti$ and $^{208}Pb$ and compared
with results obtained by: (i) shell model sum-rule techniques (ii) nuclear
matter mapped into nuclei via a local density approximation and (iii) earlier
similar calculations.
