Solvmanifolds, in particular nilmanifolds, commonly known as twisted tori, provide several examples of internal manifolds in flux compactifications towards de Sitter, Minkowski or Anti de Sitter. The relation between these manifolds and the six-dimensional torus, and the string vacua obtained on them, are the main interests of this talk. We will first present some properties of these manifolds. We will give a generic construction of their Maurer-Cartan forms out of the six-dimensional torus, via a transformation called the twist. We will then describe several Minkowski flux backgrounds of type II supergravity obtained on these manifolds. Thanks to the generalized complex geometry approach, we will show that one can obtain those solutions from solutions on the torus, via the twist transformation. The latter then acts as a solution generating technique, being able to relate backgrounds which are not T-duals. Finally, we will apply this twist transformation technique to relate Kahler/non-Kahler solutions of the heterotic string. This talk is based on a work with Enrico Goi, Ruben Minasian and Michela Petrini.