  We consider, in the electroweak standard model context, several left-right
asymmetries in $\mu e$ elastic scattering at fixed target and collider
experiments. For the former case, we show that the muon mass effects are
important in a wide energy range. We also show that these asymmetries are
sensitive to the electroweak mixing angle $\theta_W$. The effect of an extra
$Z'$ neutral vector boson appearing in a 331 model is also considered. The
capabilities of these asymmetries in the search of this extra $Z'$ are
addressed.
