Some three decades have elapsed since the phenomenon of vortex breakdown was first observed in the leading-edge vortices trailing from delta wings. Shortly after these early observations, instability, wave propagation and flow stagnation were proposed as key elements, if not explanations, for vortex breakdown. In the meantime a great deal of theoretical, numerical and experimental work has been reported, much of the latter concerned with the breakdown of vortex flows in tubes. Even so, there is still nothing approaching a complete theory of vortex breakdown, and no general agreement as to its fundamental nature. In the present review we give an overview of vortex-breakdown research to date of the current level of understanding and assess, in the light of the available experimental evidence, the various theoretical proposals which have been made over the years.