  We consider the Cauchy problem of the cubic nonlinear Schr\"odinger equation
(NLS) on $\mathbb R^d$, $d \geq 3$, with random initial data and prove almost
sure well-posedness results below the scaling critical regularity
$s_\text{crit} = \frac{d-2}{2}$. More precisely, given a function on $\mathbb
R^d$, we introduce a randomization adapted to the Wiener decomposition, and,
intrinsically, to the so-called modulation spaces. Our goal in this paper is
three-fold. (i) We prove almost sure local well-posedness of the cubic NLS
below the scaling critical regularity along with small data global existence
and scattering. (ii) We implement a probabilistic perturbation argument and
prove `conditional' almost sure global well-posedness for $d = 4$ in the
defocusing case, assuming an a priori energy bound on the critical Sobolev norm
of the nonlinear part of a solution; when $d \ne 4$, we show that conditional
almost sure global well-posedness in the defocusing case also holds under an
additional assumption of global well-posedness of solutions to the defocusing
cubic NLS with deterministic initial data in the critical Sobolev regularity.
(iii) Lastly, we prove global well-posedness and scattering with a large
probability for initial data randomized on dilated cubes.
