Abstract: To permit cutting of rotatable machine tools, or sharpening of cutting edges therefrom, for example boring tools, drills, milling cutters and the like, a grinding head has positioned thereon two spindles which carry at least three grinding wheels; one of the spindles (41) carries two grinding wheels (43, 44) and is rotatable about a horizontal axis of rotation (42). The other spindle (37) carries a grinding wheel (35) and may carry a second one at the other end (35a) and is positioned for rotation in an essentially vertical plane. The spindles can be tilted out of position in their normal rest planes by program-controlled (47) motors, such as stepping motors or servo motors. The workpiece tool (13, 50) is held in a tool carrier (7) which is rotatable with respect to the grinding head, and movable with respect thereto by cross slide positioning means (4, 5; 2, 19).