  [Abridged] The study of short-duration gamma-ray bursts (GRBs) experienced a
complete revolution in recent years thanks to the discovery of the first
afterglows and host galaxies in May 2005. These observations demonstrated that
short GRBs are cosmological in origin, reside in both star forming and
elliptical galaxies, are not associated with supernovae, and span a wide
isotropic-equivalent energy range of ~10^48-10^52 erg. However, a fundamental
question remains unanswered: What are the progenitors of short GRBs? The most
popular theoretical model invokes the coalescence of compact object binaries
with neutron star and/or black hole constituents. However, additional
possibilities 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 of
neutron stars. In this review I summarize our current knowledge of the galactic
and sub-galactic environments of short GRBs, and use these observations to draw
inferences about the progenitor population. The most crucial results are: (i)
some short GRBs explode in dead elliptical galaxies; (ii) the majority of short
GRBs occur in star forming galaxies; (iii) the star forming hosts of short GRBs
are distinct from those of long GRBs (lower star formation rates, and higher
luminosities and metallicities), and instead appear to be drawn from the
general field galaxy population; (iv) the physical offsets of short GRBs
relative to their host galaxy centers are significantly larger than for long
GRBs; (v) the observed offset distribution is in good agreement with
predictions for NS-NS binary mergers; and (vi) short GRBs trace under-luminous
locations within their hosts, but appear to be more closely correlated with the
rest-frame optical light (old stars) than the UV light (young massive stars).
