  This paper treats the dominant rational maps from the product of two very
general curves to nonsingular projective surfaces. Combining the result by
Bastianelli and Pirola, we prove that the product of two very general curves of
genus $g\geq 7$ and $g'\geq 3$ does not admit dominant rational maps of degree
$> 1$ if the image surface is non-ruled. We also treat the case of the
2-symmetric product of a curve.
