Abstract: Toric varieties are an established method for the construction of certain varieties by discrete methods. While this correspondence has lead to a fruitful interaction between discrete geometry and algebraic geometry, it is bound to rational varieties, a very limited class from a classification theoretic point of view. The purpose of the talk is to survey the vast extension of this correspondence developed in joint work with Mark Gross that also covers such classes as Calabi-Yau- or non-rational Fano varieties. The method leads to certain highly non-trivial maximal degenerations of such varieties intimately related to mirror symmetry. The degenerations also have many interesting additional features, among them the existence of generalized theta functions.