Apparatus for machining bearing rings, the apparatus comprising a part rotatable about a vertical axis of rotation, characterized by comprising: a vertically extending cavity, formed by the rotatable part, wherein the cavity has the shape of a surface of revolution around said vertical axis of rotation and provides a frustoconical upper seat, upwardly open and downwardly tapering; a lower, axial abutment means located within the cavity and having a fixed axial position along the axis of rotation; a centering sleeve, vertically slidingly accommodated in the upper frustoconical seat, wherein the centering sleeve is divided into a plurality of sectors arranged around the vertical axis of rotation of the apparatus, and wherein the sectors together define a radially inner surface of cylindrical shape, and a radially outer surface of frustoconical, downwardly tapering shape, the sleeve being capable of reaching - a vertically raised position, in which the sectors of the centering sleeve are radially spaced, to fit a bearing ring amongst them with the ring abutting against the lower axial abutment means, and - a vertically lowered position, in which the sectors are radially closer so as to center the bearing ring with respect to the vertical axis of rotation; a plurality of locking elements arranged around the upper opening of the upper seat and mounted on elements vertically sliding along and rotatable about vertical axes, whereby the locking elements are capable of reaching - a first, disengaged position, in which the locking elements are rotated away from the radially inner cylindrical surface, and - a second, engaged position, in which the locking elements are rotated so as to protrude at least partially above the radially inner cylindrical surface and are lowered along said axes; wherein the apparatus comprises first resilient means adapted for urging the centering sleeve towards its vertically lowered position. a vertically extending cavity, formed by the rotatable part, wherein the cavity has the shape of a surface of revolution around said vertical axis of rotation and provides a frustoconical upper seat, upwardly open and downwardly tapering; a lower, axial abutment means located within the cavity and having a fixed axial position along the axis of rotation; a centering sleeve, vertically slidingly accommodated in the upper frustoconical seat, wherein the centering sleeve is divided into a plurality of sectors arranged around the vertical axis of rotation of the apparatus, and wherein the sectors together define a radially inner surface of cylindrical shape, and a radially outer surface of frustoconical, downwardly tapering shape, the sleeve being capable of reaching - a vertically raised position, in which the sectors of the centering sleeve are radially spaced, to fit a bearing ring amongst them with the ring abutting against the lower axial abutment means, and - a vertically lowered position, in which the sectors are radially closer so as to center the bearing ring with respect to the vertical axis of rotation; - a vertically raised position, in which the sectors of the centering sleeve are radially spaced, to fit a bearing ring amongst them with the ring abutting against the lower axial abutment means, and - a vertically lowered position, in which the sectors are radially closer so as to center the bearing ring with respect to the vertical axis of rotation; a plurality of locking elements arranged around the upper opening of the upper seat and mounted on elements vertically sliding along and rotatable about vertical axes, whereby the locking elements are capable of reaching - a first, disengaged position, in which the locking elements are rotated away from the radially inner cylindrical surface, and - a second, engaged position, in which the locking elements are rotated so as to protrude at least partially above the radially inner cylindrical surface and are lowered along said axes; - a first, disengaged position, in which the locking elements are rotated away from the radially inner cylindrical surface, and - a second, engaged position, in which the locking elements are rotated so as to protrude at least partially above the radially inner cylindrical surface and are lowered along said axes; wherein the apparatus comprises first resilient means adapted for urging the centering sleeve towards its vertically lowered position.