Summary: We discuss the problem of enumeration of the Adam isomorphism classes of circulant digraphs. In this paper, we obtain the enumeration formula of the following isomorphism classes of circulant digraphs: (i) \(\overset\rightharpoonup{\mathcal C}(n,2)=\{\overset\rightharpoonup C(n,a_1,a_2)|\) \(a_1,a_2\not= 0\) and \((a_1,a_2,n)=1\}\). (ii) \(\overset\rightharpoonup {\mathcal C}(n,3)=\{\overset\rightharpoonup C(n,a_1,a_2,a_3)|\) \(a_i\not= 0\) and \((a_1,a_2,a_3,n)=1\), \(n=p^a\), \(p\) is prime and \(p\not= 2\}\). (iii) \(\overset\rightharpoonup {\mathcal C}(n,4)=\{\overset \rightharpoonup C(n,a_1,a_2,a_3,a_4)|\) \(a_i\not= 0\) and \((a_1,a_2,a_3,a_4,n)=1\), \(n=p^a\), \(p\) is prime and \(p\not= 2\}\).