Binary systems arise on a diverse range of scales in the universe, from twin stars to pairs of supermassive black holes. Binary systems on all scales spend at least some of their lifetime embedded in circumbinary gas, which dramatically impacts how they evolve: it affects the orbital separation and the growth of the binary components. Accretion onto the individual components can power jets, and winds that deposit substantial energy into the ISM, and broadly impact galaxy evolution. The accompanying electromagnetic signatures often are associated with gravitational wave sources. Nevertheless, the basic hydrodynamics and the long-term evolution of a binary and circumbinary disk, and the nature of the corresponding electromagnetic signatures, is still poorly understood. The goal of this program is to bring together theorists and observers working on massive black hole binaries with those working on stellar binaries. Despite the remarkable similarities of the physical problems they address, these communities rarely mix. Interest in massive black hole binaries has surged in the last few years, due to progress in numerical modeling, on-going and planned large time-domain optical surveys, and observational discoveries of circumbinary gas in young star systems. Ongoing searches for compact unresolved binaries, gravitational-wave studies with pulsar timing arrays and ultimately LISA, will continue to drive the field observationally. This program aims to initiate the next round of theoretical breakthroughs to complement upcoming observational campaigns.