  We perform a model independent analysis so as to constrain the leptoquark
(LQ) models from negative searches for $\mu \to e \gamma$, $\mu \to 3e$ decays
(and analogous processes in the $\tau$ sector), and coherent $\mu-e$ conversion
in nuclei. We considerably improve some constraints obtained by analyses known
in the literature, analyses which we show have by far underestimated the LQ
contributions to the $\mu\to 3e$. In particular we find that the coherent
$\mu-e$ conversion in nuclei mediated by the photon--conversion mechanism and
the $\mu \to 3e$ decay are golden plates where the flavor changing leptoquark
couplings, involving the second and third quark generations, can be strongly
constrained. This is due to the fact that these processes get the enhancements
by large $\log(m_q^2/m^2_{LQ})$ terms which are induced by the so-called
``photon-penguin'' diagrams. These enhancements, which produce a mild GIM
suppression in the amplitudes, have not been taken into account in the previous
analyses. We show that the $\mu \to e \gamma$ decay can set weaker constraints
on the LQ models and this is because its amplitude is strongly GIM suppressed
by the terms of order $O(m_q^2/m^2_{LQ})$. We also present the results for the
corresponding constraints in the $\tau$ sector. Finally the prospects of the
future muon experiments for the improvement of the present bounds are analyzed
and discussed.
