  We establish local well-posedness results in weak periodic function spaces
for the Cauchy problem of the Benney system. The Sobolev space $H^{1/2}\times
L^2$ is the lowest regularity attained and also we cover the energy space
$H^{1}\times L^2$, where global well-posedness follows from the conservation
laws of the system. Moreover, we show the existence of smooth explicit family
of periodic travelling waves of \emph{dnoidal} type and we prove, under certain
conditions, that this family is orbitally stable in the energy space.
