We discuss the QCD phase diagram in the presence of a strong background magnetic field. We provide numerical evidence, based on lattice simulations of QCD with $2+1$ flavours and physical quark masses, for a crossover transition at $eB=4$ GeV$^2$ (with a pseudo-critical temperature $T_c=(98\pm3)$ MeV) and for a first order phase transition at $eB=9$ GeV$^2$ (where the measured critical temperature is $T_c=(63\pm5)$ MeV). We conclude that the critical endpoint is located at $eB=eB_{CEP}$ and $T=T_{CEP}$, where $4$ GeV$^2<eB_{CEP}<9$ GeV$^2$ and $63$ MeV$<T_{CEP}<98$ MeV.