The anomalous magnetic moment of the muon \(a_\mu=(g-2)_\mu/2\) has been measured at the Brookhaven National Laboratory in 2001 and recently at the Fermilab Muon \(g-2\) Experiment. The results deviate by 4.2 \(\sigma\) from the Standard Model predictions, where the most dominant source of theoretical error comes from the Hadronic Leading Order contribution \(a_\mu^{HLO}\). Moreover, recent calculations performed on the lattice seem to be more consistent with the experimental value for \(a_\mu^{HLO}\). MUonE is a proposed experiment at CERN whose purpose is to provide a new and independent determination of \(a_\mu^{HLO}\) via elastic muon-electron scattering at low momentum transfer. To achieve a precision that is comparable to the standard timelike estimation of \(a_\mu^{HLO}\), the experiment must reach an accuracy of about 10 parts per million on the differential cross section. This requires a similar level of accuracy also from the theoretical point of view: a precise calculation of the muon-electron scattering cross section with all the relevant radiative corrections as well as quantitative estimates of all possible background processes are needed. The state of the art of the theoretical calculation for $\mu e$ scattering is presented. Then, the formulation for the next-to-next-to-leading order real and virtual lepton pair contributions are described as well as their numerical impact, obtained with the Monte Carlo event generator MESMER. These contributions are crucial to reach the precision aim of MUonE.