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Strong magnetic fields can catalyse non-perturbative quantum field theory processes which would otherwise be exponentially suppressed. In this talk, I will discuss two examples of this: production of magnetic monopoles (if they exist), and baryon number violation (within the Standard Model itself). I will present results of numerical calculations in which we found the explicit instanton and sphaleron solutions describing these processes, which demonstrate that they become unsuppressed at sufficiently strong magnetic fields. This was the basis of a recent monopole search by the MoEDAL collaboration in heavy ion collisions at the LHC, which have the strongest known magnetic fields in the Universe. The results placed new model-independent lower bounds on the mass of magnetic monopoles. I will discuss the prospects of improving these bounds and also of achieving baryon number violation in future experiments.