  Using an effective lagrangian description we analyze possible new physics
contributions to the most relevant muon number violating processes: $\mu \to e
\gamma$ and $\mu$--$e$ conversion in nuclei. We identify a general class of
models in which those processes are generated at one loop level and in which
$\mu$--$e$ conversion is enhanced with respect to $\mu \to e \gamma$ by a large
$\ln(m^2_\mu/\Lambda^2),$ where $\Lambda$ is the scale responsible for the new
physics. For this wide class of models bounds on $\mu$--$e$ conversion
constrain the scale of new physics more stringently than $\mu \to e \gamma$
already at present and, with the expected improvements in $\mu$--$e$ conversion
experiments, will push it upwards by about one order of magnitude more. To
illustrate this general result we give an explicit model containing a doubly
charged scalar and derive new bounds on its couplings to the leptons.
