A process for the production of cup tappets for reciprocating-piston machines having a pressure plate made from wear-resistant alloyed tool steel and a cup casing, welded to the pressure plate, and made from cheap and cold-extrudable steel utilizes a friction welding process to weld the two parts to one another in a crack-free manner. The cup casing blank is produced by the cold-extrusion process and is carbonized before the welding so that good welding results can thereby be achieved despite the considerable strength differences and cross-sectional differences between the pressure plate and cup casing blanks. As a result of a preheating of the pressure plate, an impairment of the cup casing blank during the frictional heating can be compensated and cracks in the workpiece during or immediately after the welding can thereby be avoided. The pressure plate blank can be designed as a flat round which can be produced by stamping and which projects radially to a pronounced extent relative to the cup casing blank, so that it can be tensioned in an overhand region during the friction welding. The outer overhang containing possible cracks is removed in a subsequent chip-removing, preferably grinding operation.