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, that has a strict analogy with its counterpart in QCD and is based upon the so-called 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.