  The main questions raised in this paper are to find the sufficient conditions
that make multi-sublinear operators $T$ and their commutators ${T_{\prod \vec b
}}$, ${T_{\sum {\vec b} }}$ to be bounded on three kinds of generalized
weighted Morrey spaces. We give the main theorems of this paper to solve the
above related questions. As corollaries of the main theorems, we give
sufficient and necessary conditions for a class of multi-sublinear operators
which are bounded on three kinds of generalized weighted Morrey spaces. As some
inportant applications, we apply the main results to the multilinear
vector-valued Calder\'on-Zygmund operators, multilinear Littlewood-Paley square
operators, multilinear pseudo-differential operators and multilinear
paraproducts.
