Machine for working the peripheral edge of glass plates, comprising: - means for advancing one or more glass plates to be worked in a substantially vertical position along a first horizontal direction X parallel to the plane of the plate through a working station comprising at least a first working head for working the plate, movable along a second substantially vertical direction Y, and including a motorized spindle which can be coupled with a grinding tool, - at least three suction-cup carrying carriages movable independently from each other along said first direction X and carrying a plurality of suction-cups which can be engaged on the plate, for advancing the glass plate along the first direction X, while precisely defining the laying plane of the plate, and - machine electronic control means for controlling said suction-cup carrying carriages and said first working head, characterized in that: - said working station comprises a second working head movable along a direction parallel to said second direction Y and also including a motorized spindle which can be coupled with a grinding tool, - the first and second working heads are arranged on opposite sides with respect to the laying plane of the glass plate, and are provided for mounting first and second drilling tools (F1, F2) on the respective motorized spindles in replacement of the grinding tools, said drilling tools being adapted to coaxially engage the glass plate from opposite sides for carrying out drilling operations on the glass plate, - the suction-cup carriages are each provided with a single aligned vertical series of suction-cups and each have a maximum dimension along the longitudinal direction X which substantially does not exceed the dimension in the longitudinal direction X of each suction-cup, - said machine electronic control means are programmable so that when there is the need of performing a drilling operation at an area of the glass plate occupied by a suction-cup carrying carriage, said carriage is brought to a different position on the plate or in a position out of the engagement on the plate while the proper position of the plate is anyhow assured by at least two of the remaining suction-cup carriages. - means for advancing one or more glass plates to be worked in a substantially vertical position along a first horizontal direction X parallel to the plane of the plate through a working station comprising at least a first working head for working the plate, movable along a second substantially vertical direction Y, and including a motorized spindle which can be coupled with a grinding tool, - at least three suction-cup carrying carriages movable independently from each other along said first direction X and carrying a plurality of suction-cups which can be engaged on the plate, for advancing the glass plate along the first direction X, while precisely defining the laying plane of the plate, and - machine electronic control means for controlling said suction-cup carrying carriages and said first working head, - said working station comprises a second working head movable along a direction parallel to said second direction Y and also including a motorized spindle which can be coupled with a grinding tool, - the first and second working heads are arranged on opposite sides with respect to the laying plane of the glass plate, and are provided for mounting first and second drilling tools (F1, F2) on the respective motorized spindles in replacement of the grinding tools, said drilling tools being adapted to coaxially engage the glass plate from opposite sides for carrying out drilling operations on the glass plate, - the suction-cup carriages are each provided with a single aligned vertical series of suction-cups and each have a maximum dimension along the longitudinal direction X which substantially does not exceed the dimension in the longitudinal direction X of each suction-cup, - said machine electronic control means are programmable so that when there is the need of performing a drilling operation at an area of the glass plate occupied by a suction-cup carrying carriage, said carriage is brought to a different position on the plate or in a position out of the engagement on the plate while the proper position of the plate is anyhow assured by at least two of the remaining suction-cup carriages.