This talk will provide an overview of recent work describing the topology of various spaces of structured configurations of vectors; for example, spaces of unit norm tight frames or spaces of normal matrices with additional constraints. Our work is largely based on the observation that these spaces arise naturally from the perspective of symplectic geometry, allowing for the application of powerful results from that field. I will explain the general ideas that we use from symplectic geometry, and illustrate them on concrete spaces of interest. I will also describe recent results on frames on manifolds — that is, structured collections of vector fields — where the topological questions become much more subtle.