Résumé: In the first part of this talk we will state and discuss the Green-Griffiths-Lang conjecture which relates different

notions of hyperbolicity for projective varieties over the complex numbers.  We will explain what it means for a projective variety to be arithmetically, analytically, or algebraically hyperbolic, and

give evidence (some old and new) for the Green-Griffiths-Lang conjecture. For instance, we will explain how to prove that a projective variety which is "hyperbolic" in any sense of the word has only finitely many automorphisms. In the second part of this talk we will formulate a non-archimedean analogue

of the Green-Griffiths-Lang conjecture (which involves a non-archimedean variant of Brody hyperbolicity). One of the main results that 

we will present is a proof of the Green-Griffiths-Lang conjecture for "constant" projective varieties defined over a non-archimedean field  K of equicharacteristic zero. The latter is joint work with Alberto Vezzani.