  In this paper, we consider the Majda-Biello system, a coupled KdV-type
system, on the torus. In the first part of the paper, it is shown that, given
initial data in a Sobolev space, the difference between the linear and the
nonlinear evolution almost always resides in a smoother space. The smoothing
index depends on number-theoretic properties of the coupling parameter in the
system which control the behavior of the resonant sets. In the second part of
the paper, we consider the forced and damped version of the system and obtain
similar smoothing estimates. These estimates are used to show the existence of
a global attractor in the energy space. We also show that when the damping is
large in relation to the forcing terms, the attractor is trivial.
