The onset of instability is one of the factors that limit the extent to which materials can be loaded or deformed. In some material systems, instability leads to localization of deformation, which after some growth, is locally arrested. However, under prevailing conditions, the local deformation can propagate and potentially destabilize the whole material domain. During the propagation of the instability, highly deformed and relatively undeformed “phases” co-exist. Furthermore, the effort required tends to remain essentially constant and lower than that needed to initiate the instability in the intact system. This presentation encompasses three material systems that exhibit this behavior known as propagating instability: the evolution of Lüders banding in mild steels, the evolution of deformation associated with phase transformation in shape memory alloys, and the propagation of crushing in honeycombs and foams. The lecture will use results from experiments and modeling to illustrate that, although the physical micromechanical reasons that cause this behavior are different in each material system, they share an underlying up-down-up local response. How this guides their constitutive modeling will be demonstrated using numerical results.