Vehicle bumper having high yield strength and extensive movement under impact loads with full shape restoration, comprising a thin transversely curved high strength resilient front plate, a high strength resilient rear backing plate located at a distance behind the front plate and anchored to the vehicle, and an intermediate body of an elastomeric material between the front plate and the rear plate to absorb impacts, increase the yield distance and to assist in restoring the front plate to original shape, together with a soft yieldable front jacket with a scuff-resistant outer surface which covers the bumper and, if desired, adjacent parts of the front of the vehicle.