In this paper we study how automated reasoning systems based on Description Logics (DLs) can be used for reasoning about UML class diagrams. The ability of reasoning automatically on UML class diagrams makes it possible to provide computer aided support during the application design phase in order to automatically detect relevant properties, such as inconsistencies and redundancies. We show that UML class diagrams can be formalized as knowledge bases expressed in the DL DLR. DLR knowledge bases can be translated into knowledge bases expressed in the variants of ALCQI accepted by state-of-the-art DL-based systems. Hence, in principle, the reasoning capabilities of such systems can be used to reason on UML class diagrams. However, we report some experiments indicating that state-of-the-art systems have still difficulty in dealing with the resulting knowledge bases.