Short gamma-ray bursts (SGRBs) are extremely bright flashes of gamma-rays, lasting less than 2 s, originating from beyond the Milky Way but their progenitors remain unknown. The most popular progenitor theory involves the merger of two compact objects, either two neutron stars (NSs) or a NS and a black hole (BH), which then collapse to form a BH. A small proportion of SGRBs may instead be giant flares from extragalactic soft gamma-ray repeaters (SGRs) in nearby galaxies. The aim of this Thesis is to place constraints on the progenitors of SGRBs using multiwavelength observational data. \ud \ud The extragalactic SGR giant flare theory is tested by considering the properties of three candidate SGRBs which may have occurred in nearby host galaxies. It is likely that only one of the three was an extragalactic SGR giant flare and, although they are all shown to be consistent with this progenitor,\ud GRB 070201 is most convincing candidate. Afterglow predictions are made for future candidates.\ud \ud Following on from the giant flare candidates, more typical SGRBs are considered. GRB 080905A is the nearest confirmed SGRB, occurring offset from a spiral galaxy at z∼0.12 which is studied using spatially resolved spectroscopy. The properties of GRB 080905A are shown to be consistent with a compact binary merger. GRB 090515 was a SGRB with an extremely unusual bright X-ray plateau and extremely steep decay phase. However, the prompt and late time properties are consistent with typical SGRBs. The plateau is explained by an unstable magnetar, formed during the SGRB, which collapses to form a BH within a few hundred seconds. The magnetar is suggested to be formed via the merger of two NSs. Many Swift SGRBs are shown to have evidence of energy injection within their X-ray lightcurves and 44–76% are consistent with forming a magnetar
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