A bumper assembly for a motor vehicle comprising a cross member that extends transverse the motor vehicle, and has an open hat-shaped hollow profile with a rear-sided web and legs, extending from the web, wherein flanges extend from a free end of the legs in the vertical direction; and an opening pointing away from the motor vehicle, in the longitudinal direction. At least one closing plate is arranged allowing the opening to be completely closed in the vertical direction and arranged allowing a partial length of less than 50% of the length of the opening to be closed in the transverse direction; and that the flanges are offset from each other with an offset in the longitudinal direction; and that the cross member has a varying cross section over its length, wherein a central region is offset upwards from the end regions with a height offset to the vertical direction.