The elastic field surrounding an arbitrarily moving screw dislocation is found, and a useful analogy with two-dimensional electromagnetic fields is pointed out. These results are applied to a screw dislocation accelerating from rest and approaching the velocity of sound asymptotically. The applied stress needed to maintain this motion is found on the assumption that the Peierls condition is satisfied near the center of the dislocation. A general integral equation of motion is derived for a simplified dislocation model, and the kind of behavior it predicts is illustrated.