A bumper having a hollow section and attached to it mounting brackets for attaching the bumper to a vehicle-in particular on longitudinal beams of a private car-is such that section walls that are spaced apart form a compression strut and a tensile strut and a pair of flanking walls joining the said struts make up the hollow section. The bracket itself is a hollow section which exhibits a length of hollow-section beam facing the longitudinal beam and a curved length which is bent out from its longitudinal axis and is for connection to the hollow section of the bumper. In addition, the hollow section of the bracket contains two side walls which are joined by transverse walls and, in the region of the curved length, one of the side walls is brought up against the other side wall forming a channel-shaped cross-section