Window cover systems include window cover material in the form of pleated panels or slats. The window cover material is suspended from a traverse track and is traversed along the track for opening and closing the window system. Arrangements for maintaining spacing and alignment of pleats or slats are provided. The alignment maintaining arrangements include traverse tapes which are substantially rigid in longitudinal and lateral directions in the plane of the tape, and are flexible in a direction perpendicular to the tape. The arrangements also include attaching the window cover material to vertical edge members and providing foldable spacer-members between adjacent edge-members. In one arrangement, a box-pleated panel of window cover fabric is suspended from a traverse track on slide-members. The slide-members are each attached to a spacer-tape at regular intervals along the spacer-tape. The spacer-tape is substantially rigid in the traverse direction and in a vertical direction perpendicular to the traverse direction. The window cover system is opened and closed by rolling and unrolling the panel and the spacer-tape around a roller located at one end of a window frame. Other arrangements include combined, tape-supported vertical slat blinds and vertical pleated drapes in which the tape(s) are supported by sprockets or wheels/pulleys.