In this talk we review the recent developments of the evaluation of the two-loop virtual correction to the electron-muon scattering, $\mu e \to\mu e$, at Next-to-Next-to-Leading order in QED. These radiative corrections are relevant for the analysis of the MUonE experiment, recently proposed at CERN. MUonE aims at the high precision determination of the QED running coupling constant in the space-like region from the measurement of the differential cross section of the elastic scattering of high-energy muons on atomic electrons. We focus our discussion on the implementation of the adaptive integrand decomposition algorithm and the interplay with available tools for the reduction and computation of multi-loop integrals. Furthermore, we comment on the progress made towards the renormalisation of the amplitude. Preliminary results are shown.