Production of magnetic monopoles in particle collisions cannot be described using perturbation theory, because of their high magnetic charge. Therefore the existing experimental results only constrain their production cross section, not the monopole mass. There are arguments that the production cross section if exponentially suppressed, in which case even light GeV-scale monopoles could be compatible with experiments. In this talk, I will discuss a different production mechanism: Schwinger pair production in strong magnetic fields. This process can be described reliably using semiclassical instanton techniques even at strong coupling. I derive direct lower bounds on the magnetic monopole mass by considering neutron stars and heavy ion collisions, which have the strongest magnetic fields in the Universe. I will also discuss open theoretical questions and prospects for future monopole searches.