  We study the birational geometry of a Fano 4-fold X from the point of view of
Mori dream spaces; more precisely, we study rational contractions of X. Here a
rational contraction is a rational map f: X-->Y, where Y is normal and
projective, which factors as a finite sequence of flips, followed by a
surjective morphism with connected fibers. Such f is called elementary if the
difference of the Picard numbers of X and Y is 1. We first give a
characterization of non-movable prime divisors in X, when X has Picard number
at least 6; this is related to the study of birational and divisorial
elementary rational contractions of X. Then we study the rational contractions
of fiber type on X which are elementary or, more generally, quasi-elementary.
The main result is that the Picard number of X is at most 11 if X has an
elementary rational contraction of fiber type, and 18 if X has a
quasi-elementary rational contraction of fiber type.
