Abstract: The invention relates to a method for grinding, polishing etc. curved surfaces of non-metallic solids, in particular glass bodies using a tool adapted at least in part to the surface to be machined and which moves relatively to the body to be machined while being compressed. This tool elastically follows the solid to be machined, and flowable grinding and/or polishing means are continuously supplied to the work space between tool and solid. The apparatus includes a rotating disk for seating and fastening the solid, elastically displacable, rigid tool presses at least against one part of the surface of the solid. Preferably it is the solid which is rotated while the tool is spatially fixed in elastically displaceable manner and is kept compressed in places against the solid.