  The MUonE experiment, that aims to provide a precise measurement of the
hadronic vacuum polarization contribution to the muon $g-2$ via elastic
muon-electron scattering, has also the potential to explore the parameter space
of light new physics. Exploiting the process $\mu^- N \to \mu^- N X$, where $N$
is the target nucleus and X is a new physics light mediator, we demonstrate
that MUonE can be sensitive to new regions of parameter space for sub-GeV dark
photons. In particular, thanks to its muon beam, MUonE will be able to explore
uncharted parameter space regions for the $L_\mu-L_\tau$ model. Finally, we
also find that MUonE can probe the parameter space of axion-like particles for
different assumptions of the couplings to electrons, muons and photons.
