Regression quantiles were introduced in Koenker & Bassett [7] as quantities of interest in developing robust estimation procedures. They can be computed by linear programming combined with post optimality techniques. Here an effective alternative is presented based on the reduced gradient algorithm for l1 fitting as described in [8] combined with a piecewise linear homotopy. There is a close connection between the two approaches (the new method essentially describes what is going on in the linear programming), but it is argued that the new approach is preferable. Its robustness as a computational procedure is illustrated by several examples which give rise to a variety of different behaviours.