How Phase of Matter Got Associated with Topology

This year’s Nobel prize in physics has been awarded partly in response to the recent theoretical and experimental advances in topological states of matter. I review the conceptual development from Thouless’ formulation of a topological invariant to the 3D topological insulator, in which the study of various seemingly unrelated problems, such as the helium-3 superfluid, semiconductor spintronics, and graphene, have all played significant roles.