  In this chapter, we give an overview of our current understanding of the
physics of accreting massive black hole binaries (MBHBs), with a special focus
on the latest developments in numerical simulations and General-Relativistic
Magnetohydrodynamics (GRMHD) simulations in particular. We give a
self-contained global picture of how to model accretion onto MBHBs, analyzing
different aspects of the system such as the dynamics of the circumbinary disk,
mini-disks, outflows, the role of magnetic fields, and electromagnetic
signatures. We discuss important questions and open problems related to these
systems, what are the advantages and disadvantages of the different numerical
approaches, and what robust knowledge we have built from simulations.
