We show that almost all circulant graphs and digraphs have automorphism groups as small as possible. Of the circulant graphs and digraphs that do not have automorphism group as small as possible, provided the smallest prime divisor of the order of the (di)graph is at least $5$, we show that almost all of them are normal circulant digraphs (that is, that the left regular representation of the cyclic groups is normal in the full automorphism group). Of the circulant graphs and digraphs that are not normal, we show that the situation is more complicated. That is, we show that there are infinite sequences of integers $S_1$, $S_2$, $S_3$ such that almost all non-normal circulant graphs and digraphs whose order is in $S_1$ are semiwreath products, almost all non-normal circulant graphs and digraphs whose order is in $S_2$ have automorphism group that of a deleted wreath product, and neither semiwreath products nor those graphs whose automorphism group is that of a deleted wreath product of circulant graphs and digraphs whose order is in $S_3$ dominates.