Since the works of Bogomolov (resp. McQuillan) on algebraic curves (resp. entire curves) in surfaces of general type with positive second Segre number, foliations are known to be an important tool in the study of the Green-Griffiths-Lang conjecture. We will show how foliations can be used to exhibit counter-examples to the approach using base loci of jet differentials. Then we will explain how to generalize these counter examples to the situation where there are no foliations. In the positive direction, we will present a new criterion ensuring that a surface with general type has a big cotangent bundle, therefore extending the results of Bogomolov and McQuillan to a larger class of surfaces of general type. These are partly joint works with S. Diverio and, independently, X. Roulleau.

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