We characterize the automorphism groups of circulant digraphs whose connection sets are relatively small, and of unit circulant digraphs. For each class, we either
explicitly determine the automorphism group or we show that the graph is \normal" circulant, so the automorphism group is contained in the normalizer of a cycle. Then we use these characterizations to prove results on the automorphisms of the endomorphism monoids of those digraphs. The paper ends with a list of open problems on graphs, number theory, groups and semigroups.