| Negativity of the holomorphic sectional curvature is a sufficient condition which insures that a compact complex manifold is Kobayashi hyperbolic, but it is far from being necessary. After reviseing the essential notions and concepts of curvature for Hermitian and/or Khler manifolds, and how they are linked with Kobayashi hyperbolicity, we shall quantify the above-mentioned lack of necessity and illustrate it also via examples. In the second part of the talk, we shall explain how it is possible, under the stronger hypothesis of negativity of the holomorphic sectional curvature, to prove a longstanding conjecture by Kobayashi stating that a compact Kobayashi hyperbolic Khler manifold is projective and canonically polarized (works of Wu-Yau, Tosatti-Yang, Diverio-Trapani, and Guenancia). |