  The Green-Griffiths-Lang conjecture stipulates that for every projective
variety X of general type over C, there exists a proper algebraic subvariety of
X containing all non constant entire curves f : C $\rightarrow$ X. Using the
formalism of directed varieties, we prove here that this assertion holds true
in case X satisfies a strong general type condition that is related to a
certain jet-semistability property of the tangent bundle TX . We then give a
sufficient criterion for the Kobayashi hyperbolicity of an arbitrary directed
variety (X,V). This work is dedicated to the memory of Professor Salah
Baouendi.
