  We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and
show that it maps polystable bundles to polystable ones. The role of ``dual''
variety to the given K3 surface $X$ is here played by a suitable component
$\hat X$ of the moduli space of stable sheaves on $X$. For a wide class of K3
surfaces $\hat X$ can be chosen to be isomorphic to $X$; then the Fourier-Mukai
transform is invertible, and the image of a zero-degree stable bundle $F$ is
stable and has the same Euler characteristic as $F$.
