Thermodynamics of magnetized matter in hot and dense QCD

This chapter, to appear in the section on QCD under extreme conditions within the Encyclopedia of Nuclear Physics, aims to provide a pedagogical introduction to the physics of quarks and gluons in the presence of high temperature, nonzero (isospin) density and strong background electromagnetic fields. Extreme conditions of these types are relevant for the description of high-energy heavy-ion collisions, neutron stars and their mergers, as well as the evolution of the early Universe in its first microsecond. Most of the existing results on this topic have been obtained by means of first-principles simulations of the discretized theory of the strong interactions, lattice Quantum Chromodynamics (QCD). This lays the focus of this review chapter, although various calculations within effective theories of QCD -- most notably chiral perturbation theory -- are also discussed. Furthermore, we provide an outlook concerning open questions and yet uncharted parameter regions within this fascinating system.