A device for the finishing of at least one lengthwise edge of a sheet of material, which at least on the one lengthwise edge has a lengthwise strip which is thinner than the remainder of the sheet of material, in that the finished lengthwise edge runs parallel to the lengthwise edge of the sheet to the side bordering the different material thicknesses, and at least two scanning devices are provided, responsive to the variations of thickness of the sheet. One scanning device is arranged to scan the lengthwise strips of the sheet which are thinner and the other scanning device is arranged to scan the sheet in its area of greater thickness. The two scanning devices are positioned transverse to the direction of movement of the sheet and are spaced so that they normally scan the sheet close to the lengthwise edge thereof, on the side bordering the different material thicknesses. If the lengthwise edge deviates to the side in one or the other direction, one or the other scanning device responds, and a drive motor is actuated correspondingly to cause the transverse adjustment of the finishing units, including the scanning devices. In this manner, the sheet is finished on the lengthwise edge exactly parallel to the adjacent lengthwise edge bordering the different material thicknesses on the side.