Although the computation of a vast class of e+ e- cross sections is fully doable in perturbative QED, the perturbative series is not well behaved unless a mechanism is devised that allows one to take into account, to all orders in the coupling constant, the effects due to multiple emissions of photons and e+ e- pairs off the incoming particles. I shall discuss one such mechanism, the collinear factorisation approach, that has a strict analogy with its counterpart in QCD and leads to the definition of "Parton" Distribution Functions (PDFs) of the electron and the positron. I shall present recent results that have improved the accuracy to which these PDFs are known, and the work in progress on the implementation of e+ e- collinear-factorisation formulae into the automated MadGraph5_aMC@NLO framework.
