A machine for processing plates, for example plates of glass, natural or synthetic stone material or slabs of wood or other materials suitable for the present technology, comprising: - a conveying device (3, 4; C) for advancing a plate to be processed in a substantially vertical position along a first horizontal X direction parallel to a general plane of the plate, into and out from a processing station, - said processing station comprising at least a first plate machining head for machining the plate, said head being arranged on one side with respect to the general plane of the plate, said machining head being movable along a second substantially vertical Y direction, and including a motorized spindle having an axis directed according to a third Z direction orthogonal to said first and second directions X, Y, said spindle being adapted to be coupled at least with a milling tool, which can be used to mill the plate along a predetermined path, so as to form an opening (A1, A2) in the plate said opening being spaced from a peripheral edge of the plate, or extending from a peripheral edge of the plate, thus resulting in the removal of a portion of the plate constituting a processing waste, and - electronic control means for controlling said machine, adapted to control said first plate machining head, characterized in that said machine comprises at least one suction-cup gripping member (V, S) for gripping said plate portion constituting the processing waste, said suction-cup gripping member (V, S) being adapted to be activated by depression and being displaceable from a retracted position to an advanced position along a direction parallel to said Z third direction, for engaging said plate portion constituting the processing waste that must be removed as a result of said milling operation, so as to hold said processing waste and prevent it from falling down after that the milling operation is completed, said gripping member (V, S) being displaceable back to its retracted position so as to remove the processing waste and hold it in a position spaced apart from the general plane of the plate, after that the milling operation is terminated. - a conveying device (3, 4; C) for advancing a plate to be processed in a substantially vertical position along a first horizontal X direction parallel to a general plane of the plate, into and out from a processing station, - said processing station comprising at least a first plate machining head for machining the plate, said head being arranged on one side with respect to the general plane of the plate, said machining head being movable along a second substantially vertical Y direction, and including a motorized spindle having an axis directed according to a third Z direction orthogonal to said first and second directions X, Y, said spindle being adapted to be coupled at least with a milling tool, which can be used to mill the plate along a predetermined path, so as to form an opening (A1, A2) in the plate said opening being spaced from a peripheral edge of the plate, or extending from a peripheral edge of the plate, thus resulting in the removal of a portion of the plate constituting a processing waste, and - electronic control means for controlling said machine, adapted to control said first plate machining head,