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 µ 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 Tc as a competition between contributions from valence quarks and sea quarks resulting in a decrease of Tc as a function of B. Finally, we discuss recent efforts to modify models in order to reproduce the behavior observed on the lattice. A. B-dependent transition temperature T0 51 B. B-dependent coupling constant 52 XI. Anisotropic pressure and magnetization 55 XII. Conclusions and outlook 57 Acknowledgments 59 A. Notation and conventions 59 B. Sum-integrals 59 C. Small and large-B expansions 61 D. Propagators in a magnetic background 61References 63 1 Another common choice is the symmetric gauge, Aµ = 1 2 (0, By, −Bx, 0).