[Abridged] The study of short-duration gamma-ray bursts (GRBs) experienced acomplete revolution in recent years thanks to the discovery of the firstafterglows and host galaxies in May 2005. These observations demonstrated thatshort GRBs are cosmological in origin, reside in both star forming andelliptical galaxies, are not associated with supernovae, and span a wideisotropic-equivalent energy range of ~10^48-10^52 erg. However, a fundamentalquestion remains unanswered: What are the progenitors of short GRBs? The mostpopular theoretical model invokes the coalescence of compact object binarieswith neutron star and/or black hole constituents. However, additionalpossibilities exist, including magnetars formed through prompt channels(massive star core-collapse) and delayed channels (binary white dwarf mergers,white dwarf accretion-induced collapse), or accretion-induced collapse ofneutron stars. In this review I summarize our current knowledge of the galacticand sub-galactic environments of short GRBs, and use these observations to drawinferences about the progenitor population. The most crucial results are: (i)some short GRBs explode in dead elliptical galaxies; (ii) the majority of shortGRBs occur in star forming galaxies; (iii) the star forming hosts of short GRBsare distinct from those of long GRBs (lower star formation rates, and higherluminosities and metallicities), and instead appear to be drawn from thegeneral field galaxy population; (iv) the physical offsets of short GRBsrelative to their host galaxy centers are significantly larger than for longGRBs; (v) the observed offset distribution is in good agreement withpredictions for NS-NS binary mergers; and (vi) short GRBs trace under-luminouslocations within their hosts, but appear to be more closely correlated with therest-frame optical light (old stars) than the UV light (young massive stars).
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