Originally, the theory of automorphic forms was concerned only with holomorphic automorphic forms on the upper half-plane or certain bounded symmetric domains. In the fifties, it was noticed (first by Gelfand and Fomin) that these automorphic forms could be viewed as smooth vectors in certain representations of the ambient group G, on spaces of functions on G invariant under the given discrete group /*. This led to the more general notion of automorphic forms on real semisimple groups, with respect to arithmetic subgroups, on adelic groups, and finally to the direct consideration of the underlying representations. The main purpose of this paper is to discuss the notions of automorphic forms on real or adelic reductive groups, of automorphic representations of adelic groups, and the relations between the two. We leave out completely the passage from automorphic forms on bounded symmetric domains to automorphic forms on groups, which has been discussed in several places (see, e.g., [2], or aIso[5],[6],[15]for modular forms).