  We prove that a smooth Fano hypersurface $V=V_M\subset{\Bbb P}^M$, $M\geq 6$,
is birationally superrigid. In particular, it cannot be fibered into uniruled
varieties by a non-trivial rational map and each birational map onto a minimal
Fano variety of the same dimension is a biregular isomorphism. The proof is
based on the method of maximal singularities combined with the connectedness
principle of Shokurov and Koll\' ar.
