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This Dissertation is focused on the evolution of massive black hole binaries embedded in gaseous accretion disks. Mergers of massive black holes across a range of mass ratios are powerful sources of gravitational waves (GWs) detectable by the future space-based detector, the Laser Interferometer Space Antenna (LISA). In many cases these sources may reside in Active Galactic Nuclei, in which they are embedded in a dense accretion disk. Interactions with surrounding gas can affect their orbital evolution, leaving signatures in both GWs and in electromagnetic emission. First, we present two-dimensional hydrodynamical simulations of accretion disks with embedded intermediate mass ratio inspirals. We demonstrate that torques from the gas disk can affect a coalescing BH, producing deviations in the GW signal. Whether or not the gas slows down or speeds up the inspiral, and whether the resultant deviation is detectable, is dependent on the system mass ratio, the disk parameters, and the evolutionary stage of the binary. With a suite of simulations varying these characteristics, we elucidate the sensitivity of the gas imprint and its detectability to mass ratio, disk viscosity, and Mach number. Since the characteristic imprint on the GW signal is strongly dependent on disk parameters, a LISA detection of a gas-embedded inspiral would probe the physics of AGN disks and migration. Finally, we explore an electromagnetic signature of a circumbinary disk produced in response to a massive black hole binary merger. With hydrodynamical simulations that resolve the vertical structure of a circumbinary disk, we show that the change in potential produced during the final coalescence of a binary can perturb the surrounding material, producing shocks above the disk midplane, and that this response depends on the disk temperature. This carries implications for the associated emission following the GW signal, which may produce non-thermal radiation that varies with disk properties.