The holomorphic sectional curvature of a Kahler metric is the Riemannian sectional curvature of holomorphic 2-planes. It determines the whole curvature tensor, but in a nontrivial way, and it has remained a rather mysterious object. I will describe some very recent progress around this topic, including the following solution of a conjecture of Yau: if a compact Kahler manifold has negative holomorphic sectional curvature, then it is projective and it admits another Kahler metric with negative Ricci curvature. This is joint work with Xiaokui Yang, building on earlier work
of Wu-Yau.