Shape change is increasingly used in physical user interfaces, both as input and output. Yet, the progress made and the key research questions for shape-changing interfaces are rarely analyzed systematically. We review a sample of existing work on shape-changing interfaces to address these shortcomings. We identify eight types of shape that are transformed in various ways to serve both functional and hedonic design purposes. Interaction with shape-changing interfaces is simple and rarely merges input and output. Three questions are discussed based on the review: (a) which design purposes may shape-changing interfaces be used for, (b) which parts of the design space are not well understood, and (c) why studying user experience with shape-changing interfaces is important.