Abstract: In this talk I will discuss new developments that illustrate the interplay between topology and geometry in topological phases of matter. I will discuss the classification of some topological insulator/superconductor phases via their spatial symmetries and the consequences for topological defects such as disclinations and dislocations. Additionally, I will show how spatial symmetries can protect quantized topological responses in topological insulator phases. Finally, I will cover the chiral gravitational response of 2+1-d time-reversal breaking Chern insulators including the topological viscosity response and the bulk-boundary correspondence. If time permits, I will discuss how the concepts developed in the discussion of disclination defects may be adapted to predict non-trivial consequences for the melting of 2D phases of matter which have spatial order coexisting with an additional order (e.g. superfluid order).