  This paper studies the birational geometry of terminal Gorenstein Fano
3-folds. If Y is not Q-factorial, in most cases, it is possible to describe
explicitly the divisor class group Cl Y by running a Minimal Model Program
(MMP) on X, a small Q-factorialisation of Y. In this case, Weil non-Cartier
divisors are generated by "topological traces " of K-negative extremal
contractions on X. One can show, as an application of these methods, that a
number of families of non-factorial terminal Gorenstein Fano 3-folds are
rational. In particular, I give some examples of rational quartic hypersurfaces
with Cl Y of rank 2, and show that when Cl Y has rank greater than 6, Y is
always rational.
