  We present an illustrative discussion of the physics potential of $\mu^-$-
$e^+$ conversion. We point out that this process, although much less frequently
studied than the related but much more popular processes of $\mu^-$- $e^-$
conversion and neutrinoless double beta decay, in fact is a promising
alternative possibility to detect both lepton flavour and number violation.
However, for this goal to be reached, a combined effort of experiments and
theory, both in nuclear and particle physics, is necessary to advance. The aim
of this paper is to be an "appetiser" to trigger such an initiative.
