In this paper, we introduce a new technique for estimating the optical flow field, starting from image sequences. As suggested by Fleet and Jepson, we track contours of constant phase over time, since these are more robust to variations in lighting conditions and deviations from pure translation than contours of constant amplitude. Our phase-based approach proceeds in three stages. First, the image sequence is spatially filtered using a bank of quadrature pairs of Gabor filters, and the temporal phase gradient is computed, yielding estimates of the velocity component in directions orthogonal to the filter pairs' orientations. Second, a component velocity is rejected if the corresponding filter pair's phase information is not linear over a given time span. Third, the remaining component velocities at a single spatial location are combined and a recurrent neural network is used to derive the full velocity. We test our approach on several image sequences, both synthetic and realistic.