The fly press comprises a flywheel means and, in combination, at least two screws, connected to the flywheel means, ram drive means for at least one ram, and at least two nuts mounted in said ram and peripherally movable and axially immovable in said ram. Overload-preventing means are arranged between said nuts and said ram and are adapted to be nonresponsive to torques below a minimum torque between said nuts and ram, so that such torques will be fully transmitted, whereas the overload-preventing means are responsive to torques which are higher than the minimum torque so that such higher torques will not be transmitted.