The existence of photon-photon scattering historically was one of the first non-trivial predictions of QED. However, since the cross section is very small at low energies it was only in 2017 that a direct measurement was achieved in heavy-ion collisions at the LHC. After short reviews of the history of the subject and of the CERN experiment, I discuss the general structure of the four-photon tensor, which also serves as the prototype for all vertices of four gauge bosons. I then come to generalizations from QED to the Standard Model and beyond, in particular to the Born-Infeld theory that has gained some popularity in recent years as an alternative to standard quantum electrodynamics.