A nanostructure die with a concavely curved nanostructured die surface for seamless embossing of at least one peripheral ring of a jacket surface of an embossing roll in a step-and-repeat process and an embossing roll for continuous embossing of nanostructures with an embossing layer, which has been applied on a body of revolution, with a jacket surface with at least one peripheral ring which is made seamless at least in the peripheral direction and which is embossed in the step-and-repeat process. Furthermore, the invention relates to a method and a device for producing such an embossing roll for continuous embossing of nanostructures as well as a method for producing such a nanostructure die and a method for producing an embossing substrate.