  A confirmation of the long-standing muon $g$-2 discrepancy requires both
experimental and theoretical progress. On the theory side, the hadronic
corrections are under close scrutiny, as they induce the leading uncertainty of
the Standard Model prediction. Recently, the MUonE experiment has been proposed
at CERN to provide a new determination of the leading hadronic contribution to
the muon $g$-2 via the measurement of the differential cross section of
muon-electron scattering. The precision expected at this experiment raises the
question whether possible new-physics (NP) could affect its measurements. We
address this issue studying possible NP signals in muon-electron collisions due
to heavy or light mediators, depending on whether their mass is higher or lower
than ${\cal O} (1{\rm GeV})$. We analyze the former in a model-independent way
via an effective field theory approach, whereas for the latter we focus on
scenarios with light scalar and vector bosons. Using existing experimental
bounds, we show that possible NP effects in muon-electron collisions are
expected to lie below MUonE's sensitivity. This result confirms and reinforces
the physics case of the MUonE proposal.
