Brody hyperbolic projective varieties over the complex numbers are extremely rich in properties. For instance, such varieties have only finitely many automorphisms, and every surjective endomorphism is actually an automorphism of finite order. Consequently, if one believes the conjectures of Green-Griffiths, Lang or Vojta, these properties should be shared by varieties which are “arithmetically” hyperbolic or “algebraically” hyperbolic. In this talk, we will see how to establish some of these properties, and thereby verify many of the predictions made by the conjectures of Green-Griffiths and Lang. I will report on joint works with Raymond van Bommel, Ljudmila Kamenova, Alberto Vezzani, and Junyi Xie.