  We prove that every smooth complete intersection X defined by s hypersurfaces
of degree d_1, ... , d_s in a projective space of dimension d_1 + ... + d_s is
birationally superrigid if 5s +1 is at most 2(d_1 + ... + d_s +
1)/sqrt{d_1...d_s}. In particular, X is non-rational and Bir(X)=Aut(X). We also
prove birational superrigidity of singular complete intersections with similar
numerical condition. These extend the results proved by Tommaso de Fernex.
