Abstract: A machine for abrading or polishing a workpiece comprises a holding surface holding the workpiece, a head member arranged along a rotation axis to rotate about the rotation axis, a working member having a surface for abrading or polishing the workpiece arranged on the head member on die rotational axis for rotation about the rotation axis with the head member, a first driving arrangement for driving a head member and the working member mounted thereon to rotate about the rotation axis, a head mounting arrangement for mounting the head member, a second driving arrangement for driving the head mounting arrangement to incline the rotation axis of the head member relative to a precession axis intersecting the rotation axis, and for moving the head member to inclined positions with the rotation axis processed about the precession axis, and a third driving arrangement for relatively moving the head mounting arrangement across the holding surface.