With the detection of the binary neutron star merger GW170817, a new era of multi-messenger astronomy started. GW170817 demonstrated that neutron star mergers are ideal laboratories for constraining the equation of state of cold supranuclear matter, to study the central engines of short GRBs, and to understand the origin and production of heavy elements. We discuss how the last milliseconds before and after the merger can be studied with full 3D numerical relativity simulations to obtain information about the emitted gravitational wave and electromagnetic signals. We present the first public database for binary neutron stars including more than 350 individual setups and explain how these simulations help to develop gravitational wave and electromagnetic models. These models allow us to constrain the unknown equation of state of cold supranuclear matter. Finally, we discuss the possibility that GW170817 or future detections could involve exotic compact objects, considering the example case of 'axion stars'. We show how future numerical relativity simulations will allow us to distinguish between different merger scenarios and may enhance our understanding of dark matter in the Universe.