Apparatus is disclosed for threading tapping holes in workpieces wherein the tapping hole is surrounded by a surface having alternating recesses and arcuate projections which are to be threaded. The apparatus has a hold-down plate one side of which is movable against the exposed side of the workpiece, a carrier which is reciprocable toward and away from the workpiece by a plunger and a spring, a tool holder which has external threads meshing with the internal threads of a nut in the hold-down plate and a helical cam groove receiving a pin of the carrier, and a fluted tap which is coaxially affixed to the holder and extends into the tapping hole when the hold-down plate abuts against the workpiece whereby the externally threaded ribs of the tap which alternate with the flutes extend into the recesses in the surface surrounding the tapping hole. The carrier is thereupon moved toward the hold-down plate to cause the tool holder to rotate whereby the tool holder moves axially and the tap cuts or rolls threads into the projections between the recesses in the surface surrounding the tapping hole. The length of the cam groove determines the extent of rotation of the tap, and such extent is selected with a view to ensure that each rib of the tap moves from a preceding recess into the next-following recess when the surface bounding one end of the cam groove terminates the movement of the carrier toward the hold-down plate. The carrier, the means for moving the carrier and the hold-down plate can constitute parts of a stamping machine.