We discuss the phase diagram of QCD in the presence of a strong background magnetic field, providing numerical evidence, based on lattice simulations of QCD with $2+1$ flavours and physical quark masses, that the QCD crossover turns into a first order phase transition for large enough magnetic field, with a critical endpoint located between $eB=4$ GeV$^2$ (where we found an analytic crossover at a pseudo-critical temperature $T_c=(98\pm3)$ MeV) and $eB=9$ GeV$^2$ (where the measured critical temperature is $T_c=(63\pm5)$ MeV).