In this paper, we introduce and study a new class containing absolutely summing multilinear mappings and polynomials, which we call multiple weakly summing multilinear mappings and polynomials. We investigate some interesting properties about multiple weakly ($p$; $q_1$, ${\cdots}$, $q_k$)-summing multilinear mappings and polynomials defined on Banach spaces: In particular, we prove a kind of Dvoretzky-Rogers' Theorem and an ideal property for multiple weakly ($p$; $q_1$, ${\cdots}$, $q_k$)-summing multilinear mappings and polynomials. We also prove that the Aron-Berner extensions of multiple weakly ($p$; $q_1$, ${\cdots}$, $q_k$)-summing multilinear mappings and polynomials are also multiple weakly ($p$; $q_1$, ${\cdots}$, $q_k$)-summing.