Existing and future gravitational-wave detectors will observe signals so precisely that they will be able to detect possible deviations from Einstein’s theory of relativity and the standard model of particle physics. To fully exploit this unique instrumental capability, fundamental advances are needed in the theoretical description of black holes, the gravitational waves they emit, their cosmic environment and physics beyond the standard model. Providing the necessary theoretical framework is the aim of the project GWSky, awarded with 12 million euros over the next six years by the European Research Council (ERC). The ERC Synergy grant involves four nodes, SISSA (Trieste), the Niels Bohr Institute (Copenhagen), the University of California (Los Angeles), and the Max Planck Institute for Gravitational Physics (Potsdam).