We review the influence of a magnetic background on the phase diagram of strong interactions and how the chiral and deconfining transitions can be affected. After some motivation, we summarize results for both transitions obtained in the framework of the linear sigma model coupled to quarks and to the Polyakov loop, and how they compare to other effective model approaches and to lattice QCD. Then we discuss the outcome of the magnetic MIT bag model that yields a behavior for the critical deconfining temperature which is compatible with recent lattice results and magnetic catalysis. The qualitative success of the magnetic MIT bag model hints to Tc being a confinement-driven quantity, and leads us to the discussion of its behavior as predicted within the 't Hooft limit of QCD, which is also in line with the most recent lattice QCD results provided that quarks behave paramagnetically. Finally, we present the first results of a perturbative calculation of the pressure in the limit of very high magnetic fields with a surprising behavior in the chiral limit.
