The aim of this paper is to investigate the boundedness of a multilinear Calderón-Zygmund integral operators T with generalized kernels and its commutator \(T_{\vec {b}}\) formed by \(\vec {b}=(b_{1},\cdots ,b_{m})\in (\textrm{BMO}(\mathbb {R}^n))^m\) on product of weighted Morrey spaces \(\mathcal {L}^{p_{1},\lambda }(\omega _1)\times \cdots \times \mathcal {L}^{p_{m},\lambda }(\omega _m)\) . Under assumption that the multiple weight \(A_{\vec {p}}\) satisfies a certain condition, the authors prove that the T is bounded from product of weighted Morrey spaces \(\mathcal {L}^{p_{1},\lambda }(\omega _1)\times \cdots \times \mathcal {L}^{p_{m},\lambda }(\omega _m)\) to spaces \(\mathcal {L}^{p,\lambda }(\nu _{\vec {\omega }})\) , and it is also bounded from product of spaces \(\mathcal {L}^{p_{1},\lambda }(\omega _1)\times \cdots \times \mathcal {L}^{p_{m},\lambda }(\omega _m)\) into weighted weak Morrey spaces \(W\mathcal {L}^{p,\lambda }(\nu _{\vec {\omega }})\) . Furthermore, the authors show that the \(T_{\vec {b}}\) is bounded from product of spaces \(\mathcal {L}^{p_{1},\lambda }(\omega _1)\times \cdots \times \mathcal {L}^{p_{m},\lambda }(\omega _m)\) to spaces \(\mathcal {L}^{p,\lambda }(\nu _{\vec {\omega }})\) , where \(\vec {\omega }=(\omega _1,\cdots ,\omega _m)\in A_{\vec {p}}\) , \(\nu _{\vec {\omega }}=\prod \limits _{j=1}^m\omega _j^{\frac{p}{p_j}}\) , \(\vec {p}=(p_1,\cdots ,p_m)\) and \(\frac{1}{p}=\frac{1}{p_1}+\cdots +\frac{1}{p_m}\) for \(1<q'<p_1,\cdots ,p_m<\infty \) .
