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 Quantum Chromodynamics (QCD). Lattice QCD has guided our understanding of magnetized quarks and gluons through landmark results on the phase diagram, equation of state, confinement mechanism, anomalous transport phenomena, and other effects. Some lattice results have led to new paradigms in the description of hot magnetized quark matter and provided insights to the high-energy particle physics community. Since the first lattice QCD simulations with background fields, this field has become an independent research direction. The review presents the current status and recent developments of this field, along with an outlook including open questions for the near future. The review covers the physics of strongly interacting matter in three major areas: compact stars, cosmology, and heavy-ion collisions. Strong magnetic fields are relevant for magnetars, cosmological phase transitions, and heavy-ion collisions. The year 2008 marked a major advancement with the discovery of the chiral magnetic effect. The review discusses the response of strongly interacting matter to background electromagnetic fields, including the impact of magnetic and electric fields on hadron properties, gluonic observables, and the QCD phase diagram. It also addresses the equation of state, transport phenomena, chirality, and topology in the presence of electromagnetic fields. The review concludes with a summary and outlook, highlighting open questions in the field.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 Quantum Chromodynamics (QCD). Lattice QCD has guided our understanding of magnetized quarks and gluons through landmark results on the phase diagram, equation of state, confinement mechanism, anomalous transport phenomena, and other effects. Some lattice results have led to new paradigms in the description of hot magnetized quark matter and provided insights to the high-energy particle physics community. Since the first lattice QCD simulations with background fields, this field has become an independent research direction. The review presents the current status and recent developments of this field, along with an outlook including open questions for the near future. The review covers the physics of strongly interacting matter in three major areas: compact stars, cosmology, and heavy-ion collisions. Strong magnetic fields are relevant for magnetars, cosmological phase transitions, and heavy-ion collisions. The year 2008 marked a major advancement with the discovery of the chiral magnetic effect. The review discusses the response of strongly interacting matter to background electromagnetic fields, including the impact of magnetic and electric fields on hadron properties, gluonic observables, and the QCD phase diagram. It also addresses the equation of state, transport phenomena, chirality, and topology in the presence of electromagnetic fields. The review concludes with a summary and outlook, highlighting open questions in the field.