  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.
