This paper presents a new Bayesian framework for motion segmentation--dividing a frame from an image sequence into layers representing different moving objects--by tracking edges between frames. Edges are found using the Canny edge detector, and the Expectation-Maximization algorithm is then used to fit motion models to these edges and also to calculate the probabilities of the edges obeying each motion model. The edges are also used to segment the image into regions of similar color. The most likely labeling for these regions is then calculated by using the edge probabilities, in association with a Markov Random Field-style prior. The identification of the relative depth ordering of the different motion layers is also determined, as an integral part of the process. An efficient implementation of this framework is presented for segmenting two motions (foreground and background) using two frames. It is then demonstrated how, by tracking the edges into further frames, the probabilities may be accumulated to provide an even more accurate and robust estimate, and segment an entire sequence. Further extensions are then presented to address the segmentation of more than two motions. Here, a hierarchical method of initializing the Expectation-Maximization algorithm is described, and it is demonstrated that the Minimum Description Length principle may be used to automatically select the best number of motion layers. The results from over 30 sequences (demonstrating both two and three motions) are presented and discussed.