  In this paper we examine different problems regarding complete intersection
varieties of high degree in a complex projective space. First we show how one
can deduce hyperbolicity for generic complete intersection of high multidegree
and high codimension from the known results on hypersurfaces. Then we prove a
existence theorem for jet differentials that generalizes a theorem of S.
Diverio. Finally, motivated by a conjecture of O. Debarre, we focus on the
positivity of the cotangent bundle of complete intersections, and prove some
results towards this conjecture; among other things, we prove that a generic
complete intersection surface of high multidegree in a projective space of
dimension at least four has ample cotangent bundle.
