Abstract: In the case of the grinding wheel (1) for grinding cutting tools and in the case of a tool grinding machine using such grinding wheels, the circumference of the grinding wheel (1) is formed as a peripheral ridge (5) with roof sides (3a, 3b) converging toward it, each having an abrasive coat (7a, 7b) (=double-cone wheel). Concave radii of a surface contour region (14) of the cutting tool can be ground with the ridge (5) and/or in each case one of the roof sides (3a, 3b) and all the convex contour regions can be ground with in each case one of the two roof sides (3a, 3b). Dressing of such a grinding wheel (1) is possible on the same machine tool that is used for grinding the tool. A grinding machine with such a double-cone wheel provides good workpiece accessibility, no problems with clamping means and machine paths that are small. The fitting of the machine with a number of wheels (for example coarse-grained for roughing and fine-grained for smoothing) is possible without any collisions occurring.