  We consider the stochastic nonlinear Schr\"odinger equations (SNLS) posed on
$d$-dimensional tori with either additive or multiplicative stochastic forcing.
In particular, for the one-dimensional cubic SNLS, we prove global
well-posedness in $L^2(\mathbb{T})$. As for other power-type nonlinearities,
namely (i) (super)quintic when $d = 1$ and (ii) (super)cubic when $d \geq 2$,
we prove local well-posedness in all scaling-subcritical Sobolev spaces and
global well-posedness in the energy space for the defocusing,
energy-subcritical problems.
