  We study smooth projective complex varieties with ample cotangent bundle. Our
main result is that in an abelian variety of dimension n, a complete
intersection of at least n/2 general hypersurfaces of sufficiently high degrees
has ample cotangent bundle. We discuss the conjecture that the analogous
statement should hold in the projective space. Finally, we present a
construction due to Bogomolov of varieties with ample cotangent bundle as
linear sections of a product of varieties with big cotangent bundle.
