Studying the behavior of the Kaehler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampere equations. I will explain how viscosity methods allow one to define and study the long term behavior of the normalized Kaehler-Ricci flow on mildly singular varieties of positive Kodaira dimension, generalizing results of Song and Tian who dealt with smooth minimal models. This is joint work with P. Eyssidieux and A. Zeriahi.