Deformations of affine toric varieties have been extensively studied by Klaus Altmann and depend on Minkowski decompositions of polytopes.
In this talk, which is based on joint work with Alessio Corti and Paul Hacking, I will explain how to construct smoothings of toric Fano 3-folds with Gorenstein singularities. Such varieties correspond to certain lattice 3-dimensional polytopes, which are called reflexive, and are involved in the Mirror Symmetry programme for Fano varieties.