One major problem in neuroscience is to understand how the brain uses neural activity to form representations of the external world. Combinatorial information in the firing patterns of neurons often reflects important features of the stimuli which generated them, particularly under the assumption of convex receptive fields. How can we efficiently extract such information? This talk will introduce some of the algebraic and topological methods currently in use for extracting combinatorial structure from neural codes, and discuss how algebraic signatures can be used to determine whether or not a code is convex.