  We characterize those bounded multilinear operators that factor through a
Hilbert space in terms of its behavior in finite sequences. This extends a
result, essentially due to S. Kwapie\'{n}, from the linear to the multilinear
setting. We prove that Hilbert-Schmidt and Lipschitz $2$-summing multilinear
operators naturally factor through a Hilbert space. It is also proved that the
class $\Gamma$ of all multilinear operators that factor through a Hilbert space
is a maximal multi-ideal; moreover, we give an explicit formulation of a
finitely generated tensor norm $\gamma$ which is in duality with $\Gamma$.
