  In this paper, we prove that in any projective manifold, the complements of
general hypersurfaces of sufficiently large degree are Kobayashi hyperbolic. We
also provide an effective lower bound on the degree. This confirms a conjecture
by S. Kobayashi in 1970. Our proof, based on the theory of jet differentials,
is obtained by reducing the problem to the construction of a particular example
with strong hyperbolicity properties. This approach relies the construction of
higher order logarithmic connections allowing us to construct logarithmic
Wronskians. These logarithmic Wronskians are the building blocks of the more
general logarithmic jet differentials we are able to construct. As a byproduct
of our proof, we prove a more general result on the orbifold hyperbolicity for
generic geometric orbifolds in the sense of Campana, with only one component
and large multiplicities. We also establish a Second Main theorem type result
for holomorphic entire curves intersecting general hypersurfaces, and we prove
the Kobayashi hyperbolicity of the cyclic cover of a general hypersurface,
again with an explicit lower bound on the degree of all these hypersurfaces.
