The strongest gravitational waves (GWs) in the universe are expected to arise from violent events such as the mergers of neutron star or black hole binaries. Decades of theoretical and experimental efforts could result in the first direct detection of GWs within the decade. Subsequent measurements should offer us an unprecedented view of strong-field gravity in action. A subset of neutron star binary mergers could be associated with a transient electromagnetic (EM) counterpart, and EM follow-up using radio, optical and high-energy facilities of these events are critical for improving our understanding of the physics underlying strong-field gravity astrophysics, and compact objects. I will discuss the astrophysical insights that we can learn from a EM+GW measurement and the challenges that lie ahead in pinpointing neutron star binary mergers using networks of GW interferometers and multiwavelength EM telescopes and arrays.