The unique characteristics of the atmospheric propagation of deep ultraviolet (UV) radiation make possible the novel capability of establishing non-line-of-sight (NLOS) optical communication links. Although NLOS UV communications (UVC) has been studied for decades, early work focused on the use of lasers and flash lamps as sources. Recent advances in device technology, including UV light-emitting diodes and solar-blind optical filters, suggest that compact low-power systems may soon be feasible, and, as a result, research into the effective use of this rapidly maturing technology has accelerated. In this paper, we review the NLOS UVC literature, examining a range of topics from channel modelling and experimentation through system analysis and prototype development. The breadth of this research not only indicates the growing interest in UVC technology but also suggests the existence of many avenues for continued exploration.