The folding roof has a fabric cover tensioned over a folding support frame. An active tension control (6) increases the tension with increasing road speed to prevent the roof shape changing. A passive tension control (7) compensates for temperature effects and for the stretching with age of the fabric. The tensioner comprises a strut (9) pulled by tension belts (13) looped over fixed supports (12). An alternate tensioner has tension cables each side of the fabric and linked to tilt mounted tension levers by a Bowden cable system. Servo motors can be used for the adjustment.