Gamma-ray bursts (GRBs) -- rare flashes of ~ MeV gamma-rays lasting from a fraction of a second to hundreds of seconds -- have long been among the most enigmatic of astrophysical transients. Observations during the past decade have led to a revolution in our understanding of these events, associating them with the birth of neutron stars and/ or black holes during either the collapse of a massive star or the merger of two compact objects (e.g., a neutron star and a black hole). GRBs are particularly interesting since NS-NS and NS-BH mergers are the primary target for km-scale gravitational wave observatories such as Advanced LIGO; GRBs are also one of the most promising astrophysical sources of very high- energy neutrinos and may produce many of the neutron-rich heavy elements in nature. In this talk, I will describe the physics of these enigmatic events and summarize outstanding problems. Combined electromagnetic and gravitational-wave observations of these sources in the coming decade have the potential to produce major advances in both astrophysics and fundamental physics (tests of General Relativity and of the equation of state of dense nuclear matter).