This paper investigates the possibility of adding machinery to description logic which allows one to define self-referential concepts. An example of such a concept is a narcissist, someone who loves himself. With domains in which the natural ontology is a graph instead of a tree, this extra expressive power is often desired (e.g., when writing an ontology about web pages or molecular structures). Our results show that one has to be very careful with such additions. We add self-reference to ALC with inverse. Then we obtain all well known difficulties of having individual concepts or nominals together with inverse relations and even worse, checking for concept consistency becomes undecidable. Most of this expressive power seems not to be needed and we can identify a useful fragment whose complexity does not exceed that of ALC.