The responses of single cortical neurons were measured as a function of the binocular disparity of dynamic random dot stereograms for a large sample of neurons (n = 787) from V1 of the awake macaque. From this sample, we selected 180 neurons whose tuning curves were strongly tuned for disparity, well sampled and well described by one-dimensional Gabor functions. The fitted parameters of the Gabor functions were used to resolve three outstanding issues in binocular stereopsis. First, we considered whether tuning curves can be meaningfully divided into discrete tuning types. Careful examination of the distributions of the Gabor parameters that determine tuning shape revealed no evidence for clustering. We conclude that a continuum of tuning types is present. Second, we investigated the mechanism of disparity encoding for V1 neurons. The shape of the disparity tuning function can be used to distinguish between position-encoding (in which disparity is encoded by an interocular shift in receptive field position) and phase-encoding (in which disparity is encoded by a difference in the receptive field profile in the 2 eyes). Both position and phase encoding were found to be common. This was confirmed by an independent assessment of disparity encoding based on the measurement of disparity sensitivity for sinusoidal luminance gratings of different spatial frequencies. The contributions of phase and position to disparity encoding were compared by estimating a population average of the rate of change in firing rate per degree of disparity. When this was calculated separately for the phase and position contributions, they were found to be closely similar. Third, we investigated the range of disparity tuning in V1 as a function of eccentricity in the parafoveal range. We find few cells which are selective for disparities greater than +/-1 degrees even at the largest eccentricity of approximately 5 degrees. The preferred disparity was correlated with the spatial scale of the tuning curve, and for most units lay within a +/-pi radians phase limit. Such a size-disparity correlation is potentially useful for the solution of the correspondence problem.