We review recent observations of short-hard gamma-ray bursts and their afterglows. The launch and successful ongoing operations of the Swift satellite, along with several localizations from the High-Energy Transient Explorer mission, have provoked a revolution in short-burst studies: first, by quickly providing high-quality positions to observers; and second, via rapid and sustained observations from the Swift satellite itself. We make a complete accounting of Swift-era short-burst

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... and proposed host galaxies, and discuss the implications of these observations for the distances, energetics and environments of short bursts, and the nature of their progenitors. We then review the physical modelling of short-burst afterglows: while the simplest afterglow models are inadequate to explain the observations, there have been several notable successes. Finally, we address the case of an unusual burst that threatens to upset the simple picture in which long bursts are due to the deaths of massive stars, and short bursts to compact-object merger events.