We recall how toric deformations can be understood via the decomposition of certain polytopes into Minkowski summands. Together with a combinatorial description of the vector space of the infinitesimal deformations T1, this gives rise to a natural understanding of the Kodaira-Spencer map. Then, we focus on two-dimensional quotient singularities and compare Q-Gorenstein deformations in the global and in the infinitesimal level. We will see that the corresponding functor is obstructed and hence will replace it by gadgets we call qG-deformations. This talk is based on joint work with János Kollár.