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 , mutual engagement members 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 to move from said engagement position to said disengagement position, said removal group being equipped with a blocking device 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 to move from said disengagement position to said engagement position, said engagement group providing a blocking device 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 , mutual engagement members 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 ; <EOS>