  This review provides a comprehensive summary of results on the physics of
strongly interacting matter in the presence of background electromagnetic
fields, obtained via numerical lattice simulations of the underlying theory,
Quantum Chromodynamics (QCD). Lattice QCD has guided our understanding of
magnetized quarks and gluons via landmark results on the phase diagram, the
equation of state, the confinemenent mechanism, anomalous transport phenomena
as well as many more fascinating effects. Some of the lattice results lead to
completely new paradigms in the description of hot magnetized quark matter and
provided useful insights to a broad high-energy particle physics community.
Since the first lattice QCD simulations with background fields, this field has
been established as an independent research direction. We present the current
status and recent developments of this field, together with an outlook
including open questions to be answered in the near future.
