While magnetic field strengths created in solid state physics laboratory experiments do not exceed 10^6 Gauss, much stronger fields can be present in compact astrophysical objects. For instance near the surface of a magnetar these can reach values as high as 10^{15} Gauss, with even higher values towards the core. It is well known that in such fields atoms become essentially two-dimensional objects, with a rich phenomenology.
Recently, it was realized that in heavy ion collisions at the Brookhaven National Lab as well as at the Large Hadron Collider at CERN even larger field strengths are being generated, for very short periods of time, in very small volumes. These magnetic fields are so strong that in their presence even some properties of strongly interacting matter will change, possibly with observable consequences.
I will describe theoretical expectations relating to matter in such environments. These have mostly been obtained by simulating quantum chromodynamics (QCD) on a spacetime lattice on supercomputers. QCD matter turns out to be diamagnetic at low temperatures and paramagnetic at high temperatures. The magnetic susceptibility can be decomposed into a Pauli spin contribution and angular momentum related Landau contributions.