A machine for machining the peripheral edge of plates of glass or plates of natural or synthetic stone material or plates of plastic material, in which the plates are to be machined in a substantially vertical position, said machine comprising: - a guide structure (3, 4, 5) for guiding a plate along a path in a first horizontal direction X parallel to the plane of the plate, while holding the plate in a substantially vertical position, - a processing station arranged along said path of the plate and comprising at least one machining head (H1, H2) for machining the plate, said machining head (H1, H2) being movable along a second substantially vertical Y direction, and said machining head (H1, H2) including a motorized spindle coupled with a grinding tool, - a transport device for advancing the plate along said path in said first direction X, - an electronic control unit, for controlling the position of said plate along said first X direction, for controlling the position of said at least one machining head (H1, H2) along said second Y direction and for controlling the rotation of said spindle, according to a programmed machining cycle, characterized in that said motorized spindle carried by said at least one machining head (H1, H2) which is movable along the second direction Y is coupled to a grinding tool consisting of a cup grinding wheel having an axis of rotation parallel to the plane of the plate, and in that said cup grinding wheel is orientable about an axis perpendicular to the plane of the plate, so that said cup grinding wheel is adapted to be arranged selectively at least in a first position, in which its axis of rotation is parallel to said second direction Y, and in a second position, in which its axis of rotation is parallel to said first direction X, whereby said cup grinding wheel is adapted to perform the machining of the upper edge or the lower edge of the plate when the cup grinding wheel is in said first position, with its axis of rotation parallel to said second direction Y, by means of a movement of the plate along said first direction X, with said at least one machining head (H1, H2) being in a stationary position along said second direction Y, while said cup grinding wheel is adapted to perform the machining of the front edge or the rear edge of the plate when the cup grinding wheel is in said second position, with its axis of rotation parallel to said first X direction, by means of a movement along said second direction Y of said at least one machining head, with the plate being in a stationary position along said first direction X. - a guide structure (3, 4, 5) for guiding a plate along a path in a first horizontal direction X parallel to the plane of the plate, while holding the plate in a substantially vertical position, - a processing station arranged along said path of the plate and comprising at least one machining head (H1, H2) for machining the plate, said machining head (H1, H2) being movable along a second substantially vertical Y direction, and said machining head (H1, H2) including a motorized spindle coupled with a grinding tool, - a transport device for advancing the plate along said path in said first direction X, - an electronic control unit, for controlling the position of said plate along said first X direction, for controlling the position of said at least one machining head (H1, H2) along said second Y direction and for controlling the rotation of said spindle, according to a programmed machining cycle,