Understanding color vision requires knowing how signals from the three classes of cone photoreceptor are combined in the cortex. We recorded from individual neurons in the primary visual cortex (V1) of awake monkeys while an automated, closed-loop system identified stimuli that differed in cone contrast but evoked the same response. We found that isoresponse surfaces for half the neurons were planar, which is consistent with linear processing. The remaining isoresponse surfaces were nonplanar. Some were cup-shaped, indicating sensitivity to only a narrow region of color space. Others were ellipsoidal, indicating sensitivity to all color directions. The major and minor axes of these nonplanar surfaces were often aligned to a set of three color directions that were previously identified in perceptual experiments. These results suggest that many V1 neurons combine cone signals nonlinearly and provide a new framework in which to decipher color processing in V1.