  We consider the nonlinear Schr\"odinger equation on the $d$-dimensional torus
$\mathbb T^d$, with the nonlinearity of polynomial type $|u|^{2\sigma}u$. For
any $\sigma \in \mathbb N$ and $s>\frac d2$ we prove that adding to this
equation a suitable stochastic forcing term there exists a unique global
solution for any initial data in $H^s(\mathbb T^d)$. The effect of the noise is
to prevent blow-up in finite time, differently from the deterministic setting.
Moreover we prove existence of invariant measures and their uniqueness under
more restrictive assumptions on the noise term.
