Abstract: All the stress components at a rapidly propagating crack-tipin an elastic perfectly-plastic material are the functions of θ only. Making use of this condition and the equations of steady-state motion, stress-strain relations and yield conditions, we obtain the general solutions in both the cases of anti-plane and in-plane strain. Applying these two general solutions to propagating Mode Ⅲ and Mode I cracks, respectively, the elastic perfectly-plastic and the perfectly-plastic fields at the rapidly propagating tips of Mode Ⅱ and Mode I cracks are derived.