Gamma-ray bursts (GRBs) are bright flashes of gamma rays coming from the cosmos. They occur roughly once per day, typically last for tens of seconds, and are the most luminous events in the universe. More than three decades after their discovery, and after pioneering advances from space and ground experiments, they still remain mysterious. The launch of the Swift and Fermi satellites in 2004 and 2008 brought in a trove of qualitatively new data. In this Review, we survey the interplay between these recent observations and the theoretical models of the prompt GRB emission and the subsequent afterglow.