In the presence of crystalline symmetry, gapped quantum many-body states of matter can acquire a number of new topological invariants. This raises a question of how to mathematically classify the invariants and their allowed values, how to extract the invariants from microscopic models, and how they encode quantized responses of the system. I will describe our recent advances in this area, which include a comprehensive understanding of these questions for the case where the symmetry group consists of U(1) charge conservation and orientation-preserving space group symmetries in two spatial dimensions. Some important results include (1) an understanding of quantized electric polarization for Chern insulators with crystalline symmetry, (2) a method of obtaining a complete set of crystalline invariants from partial rotations, and (3) an understanding of an invariant, referred to as the“discrete shift,” which depends on a choice of high symmetry point and encodes the charge response of the system to lattice disclinations. As an application, these results allow a new way of coloring Hofstadter’s famous butterfly with new topological invariants for the first time since TKNN’s 1982 work on the quantized Hall conductance.