  The Green-Griffiths-Lang conjecture stipulates that for every projective
variety $X$ of general type over ${\mathbb C}$, there exists a proper algebraic
subvariety of $X$ containing all non constant entire curves $f:{\mathbb C}\to
X$. 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
$T\_X$. We then use this fact to confirm a long-standing conjecture of
Kobayashi (1970), according to which a very general algebraic hypersurface of
dimension $n$ and degree at least $2n+2$ in the complex projective space
${\mathbb P}^{n+1}$ is hyperbolic.
