μ-e Conversion Versus μ-->e γ

We compare two most relevant loop induced muon number violating processes, \mu --> e \gamma and \mu-e conversion in nuclei, in constraining new physics. Because of much richer structure of the latter process it may be enhanced with respect to \mu --> e \gamma by a large $\ln(m^2_\mu/\Lambda^2),$ where $\Lambda$ is the scale of the new physics. After model-independent considerations we present two examples constraining the R-parity violating couplings of MSSM and the off-diagonal couplings of the doubly charged Higgs bosons from the non-observation of \mu-e conversion.