Process and apparatus for cutting a running material web in which the material web is wound onto several winding cores in succession, such that winding of the material web onto a new winding core begins after a predetermined diameter has been attained in a wound roll formed on an old winding core. The process includes moving the new winding core between a winding roll and the wound roll, such that the material web is fed onto the wound roll from the winding roll. The process further includes forming a nip between the new winding core and the winding roll, positioning at least one cutting device one of in a region between the wound roll and the nip and in a region upstream of the winding roll relative to a web run direction, and moving the at least one cutting device relative to the material web in a plane substantially parallel to the material web. In this manner, the running material web can be completely cut. The apparatus includes at least one cutting device adapted to completely cut the material web, and the at least one cutting device can be mounted for movement relative to the material web in a plane substantially parallel to the material web.