Theoretical and experimental results, some quite recent, on the instability and breakdown of concentrated vortices at high Reynolds numbers are reviewed. Wave related theories of the vortex breakdown phenomena are treated in some detail; these appear to provide a qualitative description of the response of vortex breakdown to variations in swirl or flow rate, and Benjamin's criticality classification, a wave-based concept, is consistent with experimental data. Known general criteria for the stability of instability of inviscid columnar vortices are reviewed, together with numerical studies of an inviscid vortex model that provides an excellent analytical fit to measured velocity profiles in vortices that experience breakdown. A new analysis of experimental data on vortex breakdown flows sheds light on the interplay between criticality and instability. The flows sufficiently far upstream of breakdowns to be unaffected by them are supercritical and stable, but they are generally closer to marginal instability than they are to criticality. The wakes are both subcritical and unstable. A conceptual framework for vortex breakdown, incorporating nonlinear wave theory and instability to three-dimensional disturbances, is suggested based on information derived from the experimental studies. (Author).