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.

Invited chapter for the edited book "New Frontiers in GRMHD Simulations" (Eds. C. Bambi, Y. Mizuno, S. Shashank and F. Yuan, Springer Singapore, expected in 2024)