  We study a class of translational-invariant insulators with discrete
rotational symmetry. These insulators have no spin-orbit coupling, and in some
cases have no time-reversal symmetry as well, i.e., the relevant symmetries are
purely crystalline. Nevertheless, topological phases exist which are
distinguished by their robust surface modes. Like many well-known topological
phases, their band topology is unveiled by the crystalline analog of Berry
phases, i.e., parallel transport across certain non-contractible loops in the
Brillouin zone. We also identify certain topological phases without any robust
surface modes -- they are uniquely distinguished by parallel transport along
bent loops, whose shapes are determined by the symmetry group. Our findings
have experimental implications in cold-atom systems, where the crystalline
Berry phase has been directly measured.
