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At some scale, Einstein's theory of general relativity (GR) must break down and be reconciled with quantum mechanics in a beyond-GR theory of gravity. Binary black hole (BBH) mergers probe the non-linear, highly dynamical, strong-field regime of gravity, and gravitational waves from these systems could thus contain signatures of beyond-GR physics. Indeed, null tests and model-independent tests of GR are actively performed with data from the LIGO and Virgo gravitational wave detectors. Model-dependent tests of GR, however, remain elusive as they require predictions for BBH mergers in beyond-GR theories, a feat that can only be achieved through numerical relativity, the practice of using supercomputers to solve the equations governing the behavior of spacetime. In this talk, I will describe our results in producing the first numerical beyond-GR BBH merger simulations in higher-curvature theories of gravity. I will discuss the BBH merger waveforms we have generated, the physics that we have learned from these simulations, and the goals for this new field of numerical relativity beyond general relativity in the coming decades of gravitational wave astronomy.