The existence of K\"ahler-Einstein metrics on a compact K\"ahler 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 K\"ahler-Ricci flow is the most canonical way to construct
K\"ahler-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
K\"ahler-Einstein metric can be constructed as the singular collapsing
limit by the K\"ahler-Ricci flow on minimal surfaces of Kodaira dimension
one. Some recent results of K\"ahler-Einstein metrics on K\"ahler
manifolds of positive first Chern class will also be discussed.