This dissertation describes recent theoretical and experimental efforts to understand the areas of the visual cortex and the neural coding underlying our ability to see and recognize objects. The first chapter is theoretical, describing a computer model of color processing in macaque primary visual cortex (V1). This previously published study shows that the apparent complexity of V1 neurons' responses to color grating stimuli may arise from the statistics of the natural visual world in combination with a learning rule under which neurons reduce the redundancy of their responses to natural stimuli. The second chapter is a hybrid of theoretical work and preliminary experimental data. We ask how one could begin to study the function of higher visual areas for which simple stimuli are insufficient. We suggest a "bottom-up" approach that starts with the visual area below the area of interest, and uses knowledge about the lower area to define a basis set of stimuli for studying the higher area. Since neurons in higher visual areas respond invariantly to stimuli, we introduce a computational topology method that allows us to rigorously study known invariances and indentify unanticipated invariances. The third chapter is a significant new experimental contribution, a method for defining and describing the functional maps in individual visual areas of the cortex. We adapt the continuous intrinsic signal imaging technique of Kalatsky and Stryker to characterize multiple functional maps, including ocular dominance, direction, spatial frequency, and temporal frequency, in primary visual areas 17 and 18 and extrastriate visual area 21a of the cat. We show that the functional architecture of area 21a is distinct from earlier visual areas, and that its function is closely linked with other ventral stream visual areas involved in object recognition. Finally, we provide a systematic classification, inspired by machine learning theory, of attempts to study the selectivity of neurons in object recognition areas of the visual brain, and discuss what the functional architecture of a visual area implies about its neural responses.