The phases of hot/dense/magnetized QCD from the lattice

In this talk I will discuss the phase structure of QCD in the presence of high temperatures, strong background magnetic fields and nonzero isospin asymmetry. The phase diagram in the space of these parameters can be explored via first principles lattice simulations, and contributes to our understanding of the physics of off-central heavy-ion collisions, compact stars and the evolution of the early universe. Following a short introduction, the first part of the talk will summarize the current status of research on the phase diagram with background magnetic fields. The second part will be devoted to the presentation of recent results on QCD with an isospin-asymmetry. The latter setup poses various challenges for the simulations that require conceptual improvements. These, together with potential applications of the results, will be addressed in some detail.