The formation and evolution of binary stars are critical components of several fields in astronomy. The most numerous sources for gravitational wave observatories are inspiraling or merging compact binaries, while binary stars are present in nearly every electromagnetic survey regardless of the target population. Simulations of large binary populations serve to both predict and inform observations of electromagnetic and gravitational wave sources. Binary population synthesis is a tool that balances physical modeling with simulation speed to produce large binary populations on timescales of days. We present a community-developed binary population synthesis suite, COSMIC, which is designed to simulate compact-object binary populations and their progenitors. As a proof of concept, we simulate the Galactic population of compact binaries and their gravitational wave signals observable by the Laser Interferometer Space Antenna.