Abstract: In a machine for cutting off or planing off road surfaces, having a steerable chassis with an independent drive motor, and a planing or cutting tool which is located within the contour base line of the chassis and is fixed in vertically adjustable manner on the chassis, a further planing or cutting tool of smaller dimensions is provided which is located outside the contour lines of the chassis. The further planing or cutting tool sits on a bracket which is arranged laterally in translatory moveable manner on the chassis so that the further planing or cutting tool on the bracket can be positioned in a dead space in front of, behind or above the chassis, which dead space is located within the contour lines of the chassis. The further planing or cutting tool permits the cutting of special profiles at the edge of the roadway but, on account of its capacity to be moved, does not restrict the useability of the cutting machine in traffic.