  We introduce and develop the notion of hyper-ideals of multilinear operators
between Banach spaces. While the well studied notion of ideals of multilinear
operators (multi-ideals) relies on the composition with linear operators, the
notion we propose, by considering the composition with multilinear operators,
explores more deeply the nonlinear feature of the subject. The results we prove
show that, although more restrictive a priori, hyper-ideals enjoy nice general
properties and its theory is rich enough to provide distinguished examples in
every situation where the corresponding multi-ideal fails to be a hyper-ideal.
