  We prove some results on the fibers and images of rational maps from a
hyper-K\"ahler manifold. We study in particular the minimal genus of fibers of
a fibration into curves. The last section of this paper is devoted to the study
of the rational map defined by a linear system on a hyper-K\"ahler fourfold
satisfying numerical conditions similar to those considered by O'Grady in his
study of fourfolds numerically equivalent to $K3^{[2]}$. We extend his results
to this more general context.
