A recent breakthrough of Wu and Yau asserts that a compact projective Kahler 
manifold with negative holomorphic sectional curvature must have ample 
canonical line bundle. In the talk, we will talk about some of the recent 
advances along this direction. In particular, we will discuss the case 
where the manifold is a noncompact Kahler manifold. We will also discuss 
the case when the Kahlerity is a priori unknown. Part of these are joint 
work with S. Huang, L.-F. Tam, F. Tong.