Magneto-thermodynamics of quarks and gluons on the lattice

Quarks and gluons are the elementary particles of nature that build up 99% of the visible universe. While being confined inside protons and neutrons at low energies, quarks and gluons deconfine at extreme temperatures, forming a plasma phase that is routinely produced in contemporary high-energy heavy-ion collisions. The physics of deconfined quarks and gluons is also of relevance for the evolution of the early universe and for neutron star physics. In these systems, relevant phenomena occur at nonzero temperature, density and/or background electromagnetic fields. How can we study the deconfinement of quarks and gluons based on first principles? What are the characteristic features of this phase transition and how do the above mentioned parameters affect them? In this talk I report on recent lattice simulations that aim to answer these questions.