A cutting device for cutting material covering wine bottle necks comprising cutters defining at least three cutting surfaces and a frame carrying the cutters. The frame has an axis and positions the cutters such that the cutting surfaces are radially spaced from the axis, aligned transversely in the axis and circumferentially spread with the angular displacement between each cutting surface and the two adjacent cutting surfaces less than 180°, whereby, upon rotation of the frame about the axis by an amount less than 180°, the cutting surfaces respectively define continuous arcs of a common circle. The frame further permits the cutting surfaces to so define a plurality of such sommon circles of different diameters.