Abstract: A precision high speed turning machine apparatus is provided which performs a selected operation, such as shaping, polishing or probing/measuring on a workpiece and in particular on a rotationally non-symmetric workpiece. The workpiece is mounted to a workpiece mount which rotates the workpiece at a controlled rate, and a tool is provided which is adapted to perform the selected operation. The tool is supported on a tool mount which positions the tool adjacent a portion of a workpiece surface to be operated on. The tool mount is designed to have very low inertia which permits relatively large movements, of the tool generally perpendicular to the workpiece, at high enough accelerations so as to permit high speed rotation of the workpiece. For some embodiments, more than one tool is mounted to the arm so as to permit a plurality of different operations to be performed.