  We present the first calculation of the complete set of NNLO QED corrections
for muon-electron scattering. This includes leptonic, non-perturbative
hadronic, and photonic contributions. All fermionic corrections as well as the
photonic subset that only corrects the electron or the muon line are included
with full mass dependence. The genuine four-point two-loop topologies are
computed as an expansion in the small electron mass, taking into account both,
logarithmically enhanced as well as constant mass effects using massification.
A fast and stable implementation of the numerically delicate real-virtual
contribution is achieved by combining OpenLoops with next-to-soft
stabilisation. All matrix elements are implemented in the McMule framework,
which allows for the fully-differential calculation of any infrared-safe
observable. This calculation is to be viewed in the context of the MUonE
experiment requiring a background prediction at the level of 10 ppm. Our
results thus represent a major milestone towards this ambitious precision goal.
