  This paper is devoted to the study of various aspects of deformations of log
pairs, especially in connection to questions related to the invariance of
singularities and log plurigenera. In particular, using recent results from the
minimal model program, we obtain an extension theorem for adjoint divisors in
the spirit of Siu and Kawamata and more recent works of Hacon and McKernan. Our
main motivation however comes from the study of deformations of Fano varieties.
Our first application regards the behavior of Mori chamber decompositions in
families of Fano varieties: we prove that, in the case of mild singularities,
such decomposition is rigid under deformation when the dimension is small. We
then turn to analyze deformation properties of toric Fano varieties, and prove
that every simplicial toric Fano variety with at most terminal singularities is
rigid under deformations (and in particular is not smoothable, if singular).
