In a method for manufacturing semiconductor devices and other finely structured parts, a projection objective (5) is used in order to project the image of a pattern arranged in the object plane of the projection objective onto a photosensitive substrate which is arranged in the region of the image plane (12) of the projection objective. In this case, there is set between an exit surface (15), assigned to the projection objective, for exposing light and an incoupling surface (11), assigned to the substrate, for exposing light a small finite working distance (16) which is at least temporarily smaller in size and exposure time interval than a maximum extent of an optical near field of the light emerging from the exit surface. As a result, projection objectives with very high numerical apertures in the region of NA>0.8 or more can be rendered useful for contactless projection lithography.