In Riemannian geometry, it is a basic fact that there is a strong relationship between the curvature of a Riemannian manifold on the one hand and its geometric structure on the other. A widely taught instance of this phenomenon is the Theorem of Bonnet-Myers. It essentially states that a complete connected Riemannian manifold whose Ricci curvature is bounded below by a positive constant is compact and has finite fundamental group.

Guided by the same general philosophy, we will discuss the role of various curvatures (bisectional, Ricci, holomorphic sectional, scalar, etc.) of hermitian (Kaehler) metrics in complex differential geometry. Special emphasis will be placed on very significant recent advances made in the study of the complex geometric structure of Kaehler manifolds of (semi-)definite holomorphic sectional curvature.