Martensitic phase transformations have seen significant progress in material science, although the underlying mechanisms remain a subject of ongoing debate. Topology, while extensively applied in analysing band structures of topological materials, has yet to be integrated into the explanation of martensitic phase transformations. The same goes for differential geometry, which is commonly used in the study of biological systems. This chapter serves as a prelude, providing an overview of this work. A brief introduction from the historical perspective on martensitic phase transformations and minimal surfaces, and topics in differential geometry that links these transformations with topology, is included to set the stage for the subsequent discussion on the topology of martensitic phase transformations.