Higher-order topological band structures

The interplay of topology and symmetry in a material's band structure may result in various patterns of topological states of different dimensionality on the boundary of a crystal. The protection of these ``higher-order'' boundary states comes from topology, with constraints imposed by symmetry. We review the bulk-boundary correspondence of topological crystalline band structures, which relates the topology of the bulk band structure to the pattern of the boundary states. Furthermore, recent advances in the K-theoretic classification of topological crystalline band structures are discussed.