  The exotic neutrinoless $\mu^- - e^-$ nuclear conversion is studied within
the conventional extensions of the standard model as well as in the minimal
supersymmetric (SUSY) models with and without R-parity conservation. The
dependence of the $\mu^- - e^-$ conversion rates on the nucleon and nuclear
structure is consistently taken into account. Using our calculated transition
matrix elements and the available experimental data on the branching ratio
$R_{\mu e^-}$ for $^{48}$Ti and $^{208}$Pb as well as the expected experimental
sensitivity for $^{27}$Al employed as a target in the planned at Brookhaven
$\mu^--e^-$ conversion (MECO) experiment, we extract very severe constraints
for the flavor violation parameters. We especially emphasize on the constraints
resulting for SUSY R-parity violating parameters.
