Let $\f:X \to W$ be a proper surjective map from a smooth complex projective variety $X$ to a normal variety $W$; if $\f$ has connected fibers and $-K_X$ is $\f$-ample, $\f$ is called a Fano-Mori contraction; if $\f$ is an isomorphism in codimension 2, then $\f$ is called a small contraction.

In this paper we study Fano-Mori contractions with fibers co\-ve\-red by large families of rational curves. After some general results we specialize to the case of small contractions, giving a complete description of small contractions of fivefolds with smooth fibers and relatively spanned anticanonical bundle.