We report on the analytic evaluation of the full set of two-loop four-point master integrals required for the determination of next-to-next-to-leading order corrections to $\mu e$-scattering in Quantum Electrodynamics. The results are obtained in the massless electron approximation and they retain full dependence on the muon mass. The considered integrals are also relevant for the crossing related process $e^+ e^-$ $\to \mu^+ \mu^-$, as well as for QCD corrections to the $t{\bar t}$-production at hadron colliders.