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Optical rogue waves are rare, extreme fluctuations in the value of an optical field. The term 'optical rogue wave' was first used in the context of an analogy between pulse propagation in an optical fibre and wave group propagation on deep water, but has since been generalized to describe many other processes in optics. This Review provides an overview of the field, concentrating primarily on propagation in optical fibre systems that exhibit nonlinear breather and soliton dynamics, but also discussing other optical systems in which extreme events have been reported. Although statistical features such as long-tailed probability distributions are often considered to be the defining feature of rogue waves, we emphasize the underlying physical processes that drive the appearance of extreme optical structures.