  The regular embeddings of complete bipartite graphs $K_{n,n}$ in orientable
surfaces are classified and enumerated, and their automorphism groups and
combinatorial properties are determined. The method depends on earlier
classifications in the cases where $n$ is a prime power, obtained in
collaboration with Du, Kwak, Nedela and \v{S}koviera, together with results of
It\^o, Hall, Huppert and Wielandt on factorisable groups and on finite solvable
groups.
