Explore the dynamics of binary systems and their surrounding disks in this 49-minute conference talk by Dong Lai from Cornell University. Delve into the complex interactions between circumbinary gas and embedded binary systems, ranging from stellar pairs to supermassive black holes in galactic nuclei. Examine the evolution of circumbinary gas, its influence on binary systems, and the most informative observational signatures. Learn about the latest numerical techniques used to model these systems and gain insights from the intersection of research on massive black hole binaries and stellar binaries. Presented as part of the "Building Bridges: Towards a Unified Picture of Stellar and Black Hole Binary Accretion and Evolution" conference at the Kavli Institute for Theoretical Physics, this talk contributes to bridging the gap between studies of binary systems across vastly different astrophysical scales.