  A 2-cell embedding of a graph $G$ into a closed (orientable or nonorientable)
surface is called regular if its automorphism group acts regularly on the flags
- mutually incident vertex-edge-face triples. In this paper, we classify the
regular embeddings of complete bipartite graphs $K_{n,n}$ into nonorientable
surfaces. Such regular embedding of $K_{n,n}$ exists only when $n =
2p_1^{a_1}p_2^{a_2}... p_k^{a_k}$ (a prime decomposition of $n$) and all $p_i
\equiv \pm 1 (\mod 8)$. In this case, the number of those regular embeddings of
$K_{n,n}$ up to isomorphism is $2^k$.
