  In this paper, we will obtain the strong type and weak type estimates for
vector-valued analogues of intrinsic square functions in the generalized
weighted Morrey spaces $M^{\Phi,\varphi}_{w}(\mathbb{R}^n)$. We study the
boundedness of intrinsic square functions including the Lusin area integral,
Littlewood-Paley $\mathrm{g}$-function and $\mathrm{g}_{\lambda}^{*}$ -function
and their commutators on vector-valued generalized weighted Morrey spaces
$M^{\Phi,\varphi}_{w}(l_2)$. In all the cases the conditions for the
boundedness are given either in terms of Zygmund-type integral inequalities on
$\varphi(x,r)$ without assuming any monotonicity property of $\varphi(x,r)$ on
$r$.
