Binary systems are ubiquitous in nature, from stars to the supermassive black holes expected to reside in galactic nuclei. In many circumstances, the binary system is enveloped by circumbinary gas. How does circumbinary gas evolve and influence the embedded binary? What observational signatures of this gas are most informative? What novel numerical techniques must we deploy to better model these systems? This conference will bring together two distinct communities: those working on massive black hole binaries with those working on stellar binaries. Because recent years have seen remarkable progress in observations of stellar binaries, and numerical modeling of both stars and black holes, now is the ideal time to bring together these two groups. The conference will highlight the remarkable similarities of problems at such varied astrophysical scales, and shed light on the crucial next steps for pushing the boundaries of our knowledge.