  We study birational geometry of Fano varieties, realized as double covers
$\sigma\colon V\to {\mathbb P}^M$, $M\geq 5$, branched over generic
hypersurfaces $W=W_{2(M-1)}$ of degree $2(M-1)$. We prove that the only
structures of a rationally connected fiber space on $V$ are the
pencils-subsystems of the free linear system $|-\frac12 K_V|$. The groups of
birational and biregular self-maps of the variety $V$ coincide.
