Abstract: An improved apparatus and method are disclosed for winding and wrapping rolls of web material or other objects, in which the roll (10) is wrapped prior to removal from the turret winding machine (96). A roll (16) of stretch wrap material is traversed, swiveled and tilted beside the rotating roll (10) or other object to be wrapped under the control of a traverse motor (144), swivel motor (178) and tilt motor (192) which are actuated in proportion to the speed of rotation of roll (10). For improved accuracy of wrapping, roll (16) of wrap material preferably is located along the swivel axis of the apparatus.