  We review in detail recent advances in our understanding of the phase
structure and the phase transitions of hadronic matter in strong magnetic
fields $B$ and zero quark chemical potentials $\mu_f$. Many aspects of QCD are
described using low-energy effective theories and models such as the MIT bag
model, the hadron resonance gas model, chiral perturbation theory, the
Nambu-Jona-Lasinio (NJL) model, the quark-meson (QM) model and Polyakov-loop
extended versions of the NJL and QM models. We critically examine their
properties and applications. This includes mean-field calculations as well as
approaches beyond the mean-field approximation such as the functional
renormalization group (FRG). Renormalization issues are discussed and the
influence of the vacuum fluctuations on the chiral phase transition is pointed
out. Magnetic catalysis at $T=0$ is covered as well. We discuss recent lattice
results for the thermodynamics of nonabelian gauge theories with emphasis on
$SU(2)_c$ and $SU(3)_c$. In particular, we focus on inverse magnetic catalysis
around the transition temperature $T_c$ as a competition between contributions
from valence quarks and sea quarks resulting in a decrease of $T_c$ as a
function of $B$. Finally, we discuss recent efforts to modify models in order
to reproduce the behavior observed on the lattice.
