The next two decades are expected to open the door to the first coincident detections of electromagnetic (EM) and gravitational wave (GW) signatures associated with massive black hole (MBH) binaries heading for coalescence. These detections will launch a new era of multimessenger astrophysics by expanding this growing field to the low-frequency GW regime and will provide unprecedented understanding of the evolution of MBHs and galaxies. They will also constitute fundamentally new probes of cosmology and would enable unique tests of gravity. The aim of this Living Review is to provide an introduction to this research topic by presenting a summary of key findings, physical processes and ideas pertaining to EM counterparts to MBH mergers as they are known at the time of this writing. We review current observational evidence for close MBH binaries, discuss relevant physical processes and timescales, and summarize the possible EM counterparts to GWs in the precursor, coalescence, and afterglow stages of a MBH merger. We also describe open questions and discuss future prospects in this dynamic and quick paced research area.
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