Present approaches to the semantics of the German prepositions auf an and unter draw on two propositions: First, that spatial prepositions in general specify a region in the surrounding of the relatum object. Second, that in the case of auf an and unter, these regions are to be defined with concepts like the vertical and/or the topological surfa¬ce (the whole surrounding exterior of an object). The present paper argues that the first proposition is right and that the second is wrong. That is, while it is true that prepositions specify regions, the regions specified by auf, an and unter should rather be defined in terms of everyday concepts like SURFACE, SIDE and UNDERSIDE. This idea is suggested by the fact that auf an and unter refer to different regions in different kinds of relatum objects, and that these regions are the same as the regions called surfaces, sides and undersides. Furthermore, reading and usage preferences of auf an and unter can be explained by a corresponding salience of the surfaces, sides and undersides of the relatum objects in question. All in all, therefore, a close look at the use of auf an and unter with different classes of relatum objects reveals problems for a semantic approach that draws on concepts like the vertical, while it suggests mea¬nings of these prepositions that refer to the surface, side and underside of an object.