A winding device for the covering of building openings, comprising a sheetlike, flexible cover element for at least partial covering of the at least one building opening and a winding element driven by a drive unit for adjusting a cover surface of the covering, wherein the cover element can be wound and unwound with the winding element. According to the invention, in addition to a first cover element, there is provided a second cover element), which can be arranged overlapping with the first cover element in the covering state, wherein the cover elements can be wound and unwound with a winding element.