  The Standard Model prediction for muon-electron scattering beyond leading
order requires the inclusion of QCD contributions which cannot be computed
perturbatively. At next-to- and next-to-next-to-leading order, they arise from
one- and two-loop diagrams with hadronic vacuum polarization insertions in the
photon propagator. We present their evaluation using the dispersive approach
with hadronic $e^+e^-$ annihilation data and estimate their uncertainty. We
find that these corrections are crucial for the analysis of future
high-precision muon-electron scattering data, like those of the recently
proposed MUonE experiment at CERN.
