A panoply of stimuli-sensitive polymorphic polymer assemblies has been constructed through the intentional synthesis of amphiphilic block copolymers comprising hydrophilic, stimulus-responsive, and hydrophobic blocks. Transformations among canonical micellar forms of polymer assemblies—spherical micelles, wormlike micelles, and vesicles (polymersomes)—have been demonstrated with a number of synthetic systems. This review discusses recent progress in the development and understanding of these systems with a focus on open questions about kinetics of shape change, effects of block copolymer architecture on the rate and nature of the transformation, and potential applications.