Recent technological advances allow for massive population-level recordings of neural activity, raising the hope of achieving a detailed understanding of the linkage of neurophysiology and behavior. Achieving this linkage relies on the tenet that, viewed in the right way, the mapping between neural activity and behavior preserves similarities. At the behavioral level, these similarities are captured by the topology and geometry of perceptual spaces. With this motivation, I describe some recent studies of the geometry of several perceptual spaces, including “low-level” spaces of visual features, and “higher-level” spaces dominated by semantic content. The experiments use a new, efficient psychophysical paradigm for collecting similarity judgments, and the analysis methods range from seeking Euclidean embeddings via non-metric multidimensional scaling to strategies that make minimal assumptions about the underlying geometry. With these tools, we characterize how the geometry of the spaces vary with semantic content, and the aspects of these geometries that are task-dependent.