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This talk will be devoted to the notion of weak Kähler hyperbolicity, recently introduced by F. Bei, S. Diverio, P. Eyssidieux et S. Trapani as a substitute of the notion of Kähler hyperbolicity (Gromov, 1989). I will review the reasons that naturally lead to this variant and we will see that being weakly Kähler hyperbolic is birationally invariant. We will finally give examples of weakly Kähler hyperbolic manifolds that do not admit any smooth Kähler hyperbolic model.