Abstract: The Kæhler-Ricci flow started as a small branch of the study of Hamilton’s Ricci flow, but by now is itself a vast area of research. In this talk, we provide an introduction to the study of the Kæhler-Ricci flow on compact Kæhler manifold, and some recent developments. We next briefly present the Kæhler-Ricci flow approach to the Analytic Minimal Model Program proposed by Song-Tian to tackle the classification of compact Kæhler manifolds. One is lead to develop a theory of weak solutions to the Kæhler-Ricci flow on Kæhler varieties with mild singularities.