A tailstock (1) for holding workpieces on tooling machines, in particular straightening machines, has an tailstock upper part (5) in which an axially mobile tailstock sleeve (7) is mounted. The tailstock upper part (5) is guided vertically on vertical guide columns (14) relative to a tailstock lower part (2). In its initial position, the tailstock upper part (5) rests against stop pins (17). The tailstock upper part (5) can follow an upward movement of the end of a workpiece it is holding.
