A bumper system having a bumper running transverse to the longitudinal direction of a vehicle and at least one connecting element mounted thereon for the purpose of attaching the bumper to a vehicle, in particular to a longitudinal beam of a passenger car. The connecting element is a multi-chamber extruded section of metal with longitudinal axis running in the longitudinal direction of the vehicle. The connecting element is in the form of a safety element which, under impact, absorbs impact energy by compressive deformation. For the purpose of connecting a towing device, an attachment member is provided that is situated in one of the hollow chambers of the connecting element.