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Magnetic catalysis is the enhancement of a condensate due to the presence ofan external magnetic field. Magnetic catalysis at $T=0$ is a robust phenomenonin low-energy theories and models of QCD as well as in lattice simulations. Wereview the underlying physics of magnetic catalysis from both perspectives. Thequark-meson model is used as a specific example of a model that exhibitsmagnetic catalysis. Regularization and renormalization are discussed and we payparticular attention to a consistent and correct determination of theparameters of the Lagrangian using the on-shell renormalization scheme. Astraightforward application of the quark-meson model and the NJL model leads tothe prediction that the chiral transition temperature $T_{\chi}$ is increasingas a function of the magnetic field $B$. This is in disagreement with latticeresults, which show that $T_{\chi}$ is a decreasing function of $B$,independent of the pion mass. The behavior can be understood in terms of theso-called valence and sea contributions to the quark condensate and thecompetition between them. We critically examine these ideas as well recentattempts to improve low-energy models using lattice input.

QCD phase diagram in a constant magnetic background. Inverse magnetic catalysis: where models meet the lattice