Kähler hyperbolic manifolds were introduced by Gromov around thirty years ago. In his seminal paper he showed that this class of Kähler manifolds enjoys several remarkable properties: for instance they are of general type, Kobayashi hyperbolic and their $L^2$-Hodge numbers $h_{(2)}^{p,q}$ are positive if and only if $p+q=\dim(M)$. In this talk I will report about some recent joint works with B. Claudon, S. Diverio, P. Eyssidieux and S. Trapani, where we introduced two weaker versions of Kähler hyperbolicity. I will explain the reasons behind our definitions and I will describe various geometric applications.