In 1967, Kobayashi introduced a pseudometric on complex manifolds, based on the Poincaré metric on the unitary disk. Understanding whether this pseudometric is an actual metric (which is the notion of Kobayashi hyperbolicity) turns out to encode important holomorphic data and is strictly connected to the existence of entire curves. After giving an overview of the basic properties of such a metric, I will focus on a conjecture by Kobayashi asking whether being Kobayashi hyperbolic for a projective variety implies the ampleness of the canonical divisor, explaining how the MMP suggests a strategy of a possible proof. If we have time, I will explain in detail the case of surfaces and how this is related to the existence result of rational curves on K3 surfaces.