Abstract: The existence of Kähler-Einstein metrics on a compact Kähler manifold of definite or vanishing first Chern class has been the subject of intense study over the last few decades, following Yau's solution to Calabi's conjecture. The Kähler-Ricci flow is the most canonical way to construct Kähler-Einstein metrics. We define and prove the existence of a family of new canonical metrics on projective manifolds with semi-ample canonical bundle, where the first Chern class is semi-definite. Such a generalized Kähler-Einstein metric can be constructed as the singular collapsing limit by the Kähler-Ricci flow on minimal surfaces of Kodaira dimension one. Some recent results of Kähler-Einstein metrics on Kähler manifolds of positive first Chern class will also be discussed.