The mergers of binary compact objects (black holes, neutron stars, white dwarfs) are amongst some of the most violent events in the Universe. The physics driving these events in strong field gravity are extremely complex, rich but still remain elusive. These cosmic laboratories present us now with both a challenge and an opportunity. The challenge is to explain the physics at play in strong-field gravity in Universe. The opportunity is to detect the accompanying radiation and panoply of multi-messenger particles (high energy neutrinos, cosmic rays and gravitational waves) for the first time with a suite of time-domain telescopes and experiments. In this pivotal new era of multi-messenger astronomy, the most compelling astrophysical sources are neutron star binary mergers, which should emit both in electromagnetic (EM) and gravitational waves (GW). I will first review the most recent advances in this blossoming field of EM+GW astronomy, which combines two active disciplines: time-domain astronomy and general relativity. I will discuss the promises of this new convergence by illustrating the wealth of astrophysical information that a combined EM+GW measurement would immediately bring. I will then outline the main challenge that lies ahead for this new field in pinpointing the sky location of neutron star mergers using GW detectors and EM wide-field synoptic surveys.