Receptive fields have long been a fundamental tool for understanding the computations being performed by individual neurons. Mathematically, viewing these individual fields as open sets in some space naturally gives rise to the more modern notion of a neural manifold. In this context, it is natural to apply methods from topology to study structure. In this talk, I will survey the topological ideas that underlie this connection. Then, I will discuss our recent work using this formalism to study how topologically structured information flows across neural populations, and from there to investigate learning such structures in spiking networks. No background in topology will be assumed.