Neural manifolds are a language for describing the structure of population activity in neural populations, which is intrinsically shaped by the structure of the information the population encodes. Tools from the mathematical field of algebraic topology allow us to detect and characterize this intrinsic structure within individual neural populations without reference to external correlates. However, as experimental tools increasingly allow us to study multiple brain regions at once, it has become possible to ask how such structured information flows and transforms as it moves through the brain. As these topological tools do not rely on prior knowledge or hypotheses about what is encoded, they are ideally suited to frame and answer such questions. In this talk, we will discuss recent work developing computational tools for studying how structure in neural manifolds align (or fails to), and forthcoming work on how we can use these tools to study learning in biological neural systems. No prior understanding of topology or topological methods in neuroscience will be assumed.