We describe an algorithm to constructively enumerate non-isomorphic union-closed sets and Moore sets. We confirm the number of isomorphism classes of union-closed sets and Moore sets on n ≤ 6 elements presented by other authors, and give the number of isomorphism classes of union-closed sets and Moore sets on 7 elements. Due to the enormous growth of the number of isomorphism classes, it seems unlikely that constructive enumeration for 8 or more elements will be possible in the foreseeable future.