The authors introduce the classes of strongly and fully almost summing multilinear mappings. These classes are closely associated with absolutely \((p;q_1,\dots, q_n)\)-summing, almost \((p_1,\dots,p_n)\)-summing, and fully \((p;q_1,\dots, q_n)\)-summing mappings, which have been explored by numerous authors. Inclusions between these classes and structural properties such as a Dvoretzky--Rogers type theorem are investigated. The type and cotype of the target space loom large in this context. Moreover, the paper includes a generalization of a result of \textit{S.\,Kwapień} [Stud.\ Math.\ 38, 193--201 (1970; Zbl 0211.43505)] which asserts that, under certain assumptions, a linear operator \(T\) is absolutely (1;1)-summing if \(T^\ast\) is absolutely \((q;q)\)-summing for some \(q \geq 1\). The proof of Proposition 1 contains a little mistake, but it can easily be corrected.