Two types of gamma-ray bursts (GRBs) are observed: short-duration hard spectrum GRBs and long-duration soft spectrum GRBs. For many years long GRBs were the focus of intense research while the lack of observational data limited the study of short-hard GRBs (SHBs). In 2005 a breakthrough occurred following the first detections of SHB afterglows, longer wavelength emission that follows the burst of gamma-rays. Similarly to long GRBs, afterglow detections led to the identification of SHB host galaxies and measurement of their redshifts. These observations established that SHBs are cosmological relativistic sources that, unlike long GRBs, do not originate from the collapse of massive stars, and therefore constitute a distinct physical phenomenon. One viable model for SHB origin is the coalescence of compact binary systems (double neutron stars or a neutron star and a black hole), in which case SHBs are the electromagnetic counterparts of strong gravitational-wave sources. The theoretical and observational study of SHBs following the recent pivotal discoveries is reviewed, along with new theoretical results that are presented here for the first time.