The MUonE experiment aims at providing a new independent evaluation of the leading hadronic contribution to the muon anomalous magnetic moment. The proposed method is based on the measurement of the running of the QED coupling in the space-like momentum region, through the elastic scattering of high-energy muons on atomic electrons. In this proceeding, the status of the theoretical effort for the MUonE experiment is reviewed. In order to achieve a competitive determination, the differential cross section of muon-electron scattering must be computed with an unprecedented precision of 10 ppm, which requires at least next-to-next-leading order corrections and the resummation of large logarithms. In addition, the simulation of the experiment needs an exhaustive theoretical description of the possible background processes, such as the lepton pair production in muon-nucleus scattering. All these processes must be implemented in a fully exclusive Monte Carlo event generator in order to be used in the MUonE data analysis.