  In this paper we study absolutely summing mappings on Banach spaces by
exploring the cotype of their domains and ranges. It is proved that every $n$%
-linear mapping from $\mathcal{L}_{\infty}$-spaces into $\mathbb{K}$ is $%
(2;2,...,2,\infty)$-summing and also shown that every $n$-linear mapping from
$\mathcal{L}_{\infty}$-spaces into $F$ is $(q;2,...,2)$-summing whenever $F$
has cotype $q.$ We also give new examples of analytic summing mappings and
polynomial and multilinear versions of a linear Extrapolation Theorem.
