  Magnetic catalysis is the enhancement of a condensate due to the presence of
an external magnetic field. Magnetic catalysis at $T=0$ is a robust phenomenon
in low-energy theories and models of QCD as well as in lattice simulations. We
review the underlying physics of magnetic catalysis from both perspectives. The
quark-meson model is used as a specific example of a model that exhibits
magnetic catalysis. Regularization and renormalization are discussed and we pay
particular attention to a consistent and correct determination of the
parameters of the Lagrangian using the on-shell renormalization scheme. A
straightforward application of the quark-meson model and the NJL model leads to
the prediction that the chiral transition temperature $T_{\chi}$ is increasing
as a function of the magnetic field $B$. This is in disagreement with lattice
results, which show that $T_{\chi}$ is a decreasing function of $B$,
independent of the pion mass. The behavior can be understood in terms of the
so-called valence and sea contributions to the quark condensate and the
competition between them. We critically examine these ideas as well recent
attempts to improve low-energy models using lattice input.
