Providing normative explanations for the structure of neural representations of sensory stimuli is a longstanding goal of theoretical neuroscience. One influential framework is the efficient coding hypothesis, which posits that biological systems aim to encode as much information as possible, subject to constraints imposed by their physical and metabolic implementation. While this theory achieved significant success in explaining representations in early visual areas, it was eventually surpassed by task-trained deep neural networks as the leading models of higher-level cortical representations. In this thesis, we develop a modern formulation of the efficient coding hypothesis by introducing an objective based on manifold capacity, a measure of how many manifolds can be reliably separated by a given representation. We show that this framework is: (1) rich enough to produce representations that support sophisticated behavior and predict highlevel cortical responses, (2) flexible enough to incorporate constraints and findings from visual and perceptual neuroscience, and (3) compatible with biologically plausible optimization schemes and canonical neural computations.