  We present an overview of recent numerical advances in the theoretical
characterization of massive binary black hole (MBBH) mergers in astrophysical
environments. These systems are among the loudest sources of gravitational
waves (GWs) in the universe and particularly promising candidates for
multimessenger astronomy. Coincident detection of GWs and electromagnetic (EM)
signals from merging MBBHs is at the frontier of contemporary astrophysics. One
major challenge in observational efforts searching for these systems is the
scarcity of strong predictions for EM signals arising before, during, and after
merger. Therefore, a great effort in theoretical work to-date has been to
characterize EM counterparts emerging from MBBHs concurrently to the GW signal,
aiming to determine distinctive observational features that will guide and
assist EM observations. To produce sharp EM predictions of MBBH mergers it is
key to model the binary inspiral down to coalescence in a full general
relativistic fashion by solving Einstein's field equations coupled with the
magnetohydrodynamics equations that govern the evolution of the accreting
plasma in strong-gravity. We review the general relativistic numerical
investigations that have explored the astrophysical manifestations of MBBH
mergers in different environments and focused on predicting potentially
observable smoking-gun EM signatures that accompany the gravitational signal.
