Excitons in quantum materials are shaped not only by electron-hole interactions, but also by the geometry and topology of the underlying electronic bands. I will discuss two systems in which these effects interplay: free excitons and defect bound excitons in two band insulators. I will present a framework for describing exciton band geometry and topology that is based on a novel approach and uses singular connections instead of the traditional Berry connection. This framework clarifies the topological contribution of interactions to the excitonic structure and identifies the singular connection as a useful tool for analyzing the topology of bound quantum states. For defect-bound excitons in topological materials, I will show that topological in-gap defect states, which have distinct features and increase robustness, lead to changes in the excitonic wave functions and binding energies as compared with trivial in gap states. Together, these results highlight excitons as sensitive probes of quantum geometry and topology, and suggest new routes for exploring the geometric and topological properties of bound states in quantum materials.