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The invention relates to a method and an apparatus for treating the edge of a piece, such as for edge grinding or the like, which is carried out in a treatment center (I) preferably by a machining means (1) in connection with a data processing unit. The machining device (1) comprises at least one machining means (1a), such as an abrasive disc or the like, treating the edge of a piece (3)on a support suface (2) in the treatment center (I). The treatment of the edge of the piece (3), supported at least for the time of the treatment phase on the support suface (2) in a stationary manner by means of a fixing arrangement (4), is carried out by the machining means (1a) preferably movable along the edge of the piece (3) and revolving, when pressed against the edge of the piece (3) to be treated, around an axis of rotation substantially perpendicular to the plane of the piece (3) to be treated. At least for facilitating the positioning of the machining means (1a) in relation to the edge to be treated, the machining means (1a) of the machining device (1) is supported at least for the time of the treatment of the piece (3), in a direction substantially perpendicular to the plane of the piece (3) to be treated, against a contact surface which is preferably solid and substantially parallel to the piece (3) and/or the support surface (2).