We develop a method for learning the spatial statistics of optical flow fields from a novel training database. Training flow fields are constructed using range images of natural scenes and 3D camera motions recovered from hand-held and car-mounted video sequences. A detailed analysis of optical flow statistics in natural scenes is presented and machine learning methods are developed to learn a Markov random field model of optical flow. The prior probability of a flow field is formulated as a Field-of-Experts model that captures the higher order spatial statistics in overlapping patches and is trained using contrastive divergence. This new optical flow prior is compared with previous robust priors and is incorporated into a recent, accurate algorithm for dense optical flow computation. Experiments with natural and synthetic sequences illustrate how the learned optical flow prior quantitatively improves flow accuracy and how it captures the rich spatial structure found in natural scene motion.