An apparatus for carrying out the replacement of an abrasive element in a machine for finishing surfaces, said apparatus in said machine comprising: - a support body configured to reversibly engage said abrasive element, said support body and said abrasive element providing, at respective engagement surfaces (15,25), mutual engagement members (16,26) of removable type configured to move from an engagement position to a disengagement position by rotating about a rotational axis in a first sense of rotation and, from said disengagement position to said engagement position, by rotating about said rotational axis in a second sense of rotation opposite to the first sense of rotation; - a displacement device configured to move said support body in the space according to at least 1 degree of freedom; said apparatus comprising at least a working station selected from the group comprising of: - a removal station in which a removal group is provided arranged to operate in combination with said displacement device for removing said abrasive element from said support body, - an application station in which an engagement group is provided arranged to operate in combination with said displacement device for applying an abrasive element to said support body; said apparatus being characterised in that said removal group and said displacement device being configured to cause a relative rotation about said rotational axis in said first sense of rotation between said engagement surface of said support body and said engagement surface of said abrasive element, in such a way to cause said mutual engagement members (16,26) to move from said engagement position to said disengagement position, said removal group being equipped with a blocking device (60,60') arranged to block one of said support body and said abrasive element in a predetermined position during said relative rotation in said first sense of rotation; said engagement group and said displacement device being configured to cause a relative rotation about said rotational axis in said second sense of rotation between said engagement surface of said support body and said engagement surface of said abrasive element, in such a way to cause said mutual engagement members (16,26) to move from said disengagement position to said engagement position, said engagement group providing a blocking device (160,160') arranged to block one of said support body and said abrasive element in a predetermined position during said rotation in said second sense of rotation. - a support body configured to reversibly engage said abrasive element, said support body and said abrasive element providing, at respective engagement surfaces (15,25), mutual engagement members (16,26) of removable type configured to move from an engagement position to a disengagement position by rotating about a rotational axis in a first sense of rotation and, from said disengagement position to said engagement position, by rotating about said rotational axis in a second sense of rotation opposite to the first sense of rotation; - a displacement device configured to move said support body in the space according to at least 1 degree of freedom; - a removal station in which a removal group is provided arranged to operate in combination with said displacement device for removing said abrasive element from said support body, - an application station in which an engagement group is provided arranged to operate in combination with said displacement device for applying an abrasive element to said support body;