Process for the beam processing of a plate-shaped or tubular workpiece, comprising a) at least one cutting procedure for producing a cutting gap along a cutting line which extends at least partially along the contour of a workpiece part to be produced from the workpiece, the cutting procedure comprising: moving a beam head for guiding a processing beam above the workpiece, wherein the processing beam is guided along the cutting line from a first cutting position to a second cutting position, and wherein the processing beam has a first power density which is dimensioned such that the workpiece is cut through, wherein the workpiece part is completely cut out of the workpiece and a residual grid of the workpiece remains, b) after complete cutting out of the workpiece part, at least one finishing procedure for finishing the residual grid along at least a part of the cutting gap, wherein the finishing procedure comprises: moving the beam head over the residual grid, wherein the processing beam is guided along a finishing line from a first finishing position to a second finishing position, wherein the processing beam has a second power density which is such that the residual grid is not cut through, and wherein the residual grid is irradiated by the processing beam along at least a part of the cutting gap in a region containing a residual grid-side cutting edge of the cutting gap, characterized in that one or more of the following finishing operations are carried out in the at least one finishing procedure: i) removing an oxide layer from the residual grid-side cutting edge of the cutting gap, ii) removing burr from the residual grid-side cutting edge of the cutting gap, iii) rounding the residual grid-side cutting edge of the cutting gap, iv) changing the shape, in particular smoothing or roughening, of the residual grid-side cutting edge of the cutting gap, v) creating a chamfer along the cutting gap, vi) coating the residual grid along the cutting gap in a region containing the residual-grid-side cutting edge of the cutting gap with a substance contained in a working gas beam. a) at least one cutting procedure for producing a cutting gap along a cutting line which extends at least partially along the contour of a workpiece part to be produced from the workpiece, the cutting procedure comprising: moving a beam head for guiding a processing beam above the workpiece, wherein the processing beam is guided along the cutting line from a first cutting position to a second cutting position, and wherein the processing beam has a first power density which is dimensioned such that the workpiece is cut through, wherein the workpiece part is completely cut out of the workpiece and a residual grid of the workpiece remains, b) after complete cutting out of the workpiece part, at least one finishing procedure for finishing the residual grid along at least a part of the cutting gap, wherein the finishing procedure comprises: moving the beam head over the residual grid, wherein the processing beam is guided along a finishing line from a first finishing position to a second finishing position, wherein the processing beam has a second power density which is such that the residual grid is not cut through, and wherein the residual grid is irradiated by the processing beam along at least a part of the cutting gap in a region containing a residual grid-side cutting edge of the cutting gap, i) removing an oxide layer from the residual grid-side cutting edge of the cutting gap, ii) removing burr from the residual grid-side cutting edge of the cutting gap, iii) rounding the residual grid-side cutting edge of the cutting gap, iv) changing the shape, in particular smoothing or roughening, of the residual grid-side cutting edge of the cutting gap, v) creating a chamfer along the cutting gap, vi) coating the residual grid along the cutting gap in a region containing the residual-grid-side cutting edge of the cutting gap with a substance contained in a working gas beam.