Solid-solid phase transitions are physical phenomena for which, due to significant variations in temperature or pressure, a solid can modify its crystal structure without undergoing any intermediate liquid phase (e.g. shape-memory alloys or the nucleation of different ice forms at elevated pressure). A key question in this setting is understanding how diffuse-interface models relate to their sharp-interface counterparts. In this talk, we introduce the variational framework commonly used to model these phenomena and discuss recent advancements in the theory of solid-solid phase transitions.