Cutting device for cutting narrow strips from a material web, in particular a rubberized material web, into or on which material web magnetically couplable elements are embedded or arranged at defined spacings one behind the other in the web longitudinal direction, comprising a blade table which is arranged on a device framework and on which the material web to be cut is able to be positioned, a cutting apparatus which comprises a cutting blade which is movable from an initial position into a cutting position relative to the blade table, and a gripping device (5, A, B) which receives and transports a cut strip and which is movable relative to the blade table, wherein - either the gripping device (5, A, B) comprises a hold-down portion which, for the purpose of cutting, is able to be placed onto the material web positioned on the blade table and which fixes the material web to the blade table during the cutting, and a magnet element which magnetically interacts with the magnetically couplable element of the cut strip for the purpose of fixing the strip to the gripping device (5, A, B) for transport into a discharge position, and wherein the hold-down portion has a cutout through which the cutting blade is movable to the blade table into the cutting position, - or the gripping device (5, A, B) comprises a hold-down portion which, for the purpose of cutting, is able to be placed onto the material web positioned on the blade table and which fixes the material web to the blade table during the cutting, and a magnet element which magnetically interacts with the magnetically couplable element of the cut strip for the purpose of fixing the strip to the gripping device (5, A, B) for transport into a discharge position, and wherein provision is made of at least one further hold-down element which, for the purpose of cutting, is able to be placed onto the material web so as to be spaced apart from the hold-down portion with formation of a clearance, wherein the cutting blade is movable to the blade table into the cutting position through the clearance. - either the gripping device (5, A, B) comprises a hold-down portion which, for the purpose of cutting, is able to be placed onto the material web positioned on the blade table and which fixes the material web to the blade table during the cutting, and a magnet element which magnetically interacts with the magnetically couplable element of the cut strip for the purpose of fixing the strip to the gripping device (5, A, B) for transport into a discharge position, and wherein the hold-down portion has a cutout through which the cutting blade is movable to the blade table into the cutting position, - or the gripping device (5, A, B) comprises a hold-down portion which, for the purpose of cutting, is able to be placed onto the material web positioned on the blade table and which fixes the material web to the blade table during the cutting, and a magnet element which magnetically interacts with the magnetically couplable element of the cut strip for the purpose of fixing the strip to the gripping device (5, A, B) for transport into a discharge position, and wherein provision is made of at least one further hold-down element which, for the purpose of cutting, is able to be placed onto the material web so as to be spaced apart from the hold-down portion with formation of a clearance, wherein the cutting blade is movable to the blade table into the cutting position through the clearance.