A shutter device co-operating with driving means for enabling the shutter (1) to be displaced between an open position and a closed position, this shutter (1) being guided by a guide track (4, 5) and being designed to close off a bay (6) or other opening. The device having a drum (8) about which the shutter (1) is wound when the latter is in the open position such that the shutter forms a roll (31) made up of turns formed by successive layers of the shutter (1). The shutter (1) can be unwound from this roll towards its closed position. The device further has confining means for preventing the turns from forming wrinkles as the shutter (1) is being displaced into the closed position and/or for preventing these turns from slipping relative to one another.