  We determine the rationality of very general quasismooth Fano 3-fold weighted
hypersurfaces completely and determine the stable rationality of them except
for cubic 3-folds. More precisely we prove that (i) very general Fano 3-fold
weighted hypersurfaces of index 1 or 2 are not stably rational except possibly
for the cubic threefolds, (ii) among the 27 families of Fano 3-fold weighted
hypersurfaces of index greater than 2, very general members of specific 7
families are not stably rational and the remaining 20 families consists of
rational varieties.
