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This thesis provides a pedagogical overview of the theoretical foundations of the McMule framework, a Monte Carlo integrator for processes with muons and other leptons. Among other things, we show how the simple infrared structure in QED can be exploited to construct FKS$^\ell$, a subtraction scheme for soft singularities to all orders in perturbation theory. Furthermore, we present the method of massification as a solution to the problem of multi-scale integrals in the presence of large scale hierarchies. Finally, we introduce next-to-soft stabilisation as an elegant tool to stabilise the numerically delicate real-virtual contribution. To this end, we generalise the Low-Burnett-Kroll theorem for massive fermions to one loop. This allows for a straightforward application of the method without the need of explicit calculations. We have developed all of these techniques with fully differential NNLO QED calculations in mind and have successfully applied them to many processes such as the muon decay as well as Bhabha and Møller scattering. One of the main drivers of these developments has been the MUonE experiment requiring a high-precision theory prediction for muon-electron ($μ$-$e$) scattering at the level of $10\, \text{ppm}$. The multi-scale nature of $μ$-$e$ scattering makes this process particularly challenging from a technical point of view. Only the combined application of FKS$^\ell$, massification, and next-to-soft stabilisation makes the corresponding calculation possible. This thesis therefore presents for the first time the fully differential calculation of the complete set of NNLO corrections to $μ$-$e$ scattering. This represents a major step towards the ambitious $10\, \text{ppm}$ target precision of the MUonE experiment.