  The cohomological rigidity problem for toric manifolds asks whether toric
manifolds are diffeomorphic (or homeomorphic) if their integral cohomology
rings are isomorphic. Many affirmative partial solutions to the problem have
been obtained and no counterexample is known. In this paper, we study the
diffeomorphism classification of toric Fano $d$-folds with $d=3,4$ or with
Picard number $\ge 2d-2$. In particular, we show that those manifolds except
for two toric Fano $4$-folds are diffeomorphic if their integral cohomology
rings are isomorphic. The exceptional two toric Fano $4$-folds (their ID
numbers are 50 and 57 on a list of {\O}bro) have isomorphic cohomology rings
and their total Pontryagin classes are preserved under an isomorphism between
their cohomology rings, but we do not know whether they are diffeomorphic or
homeomorphic.
