  We investigate the summability of the coefficients of $m$-homogeneous
polynomials and $m$-linear mappings defined on $\ell_{p}$-spaces. In our
research we obtain results on the summability of the coefficients of $m$-linear
mappings defined on $\ell_{p_{1}} \times \cdots \times \ell_{p_{m}}$. The first
results in this respect go back to Littlewood and Bohnenblust and Hille (for
bilinear and $m$-linear forms on $c_{0}$) and Hardy and Littlewood and
Praciano-Pereira (for bilinear and $m$-linear forms on arbitrary
$\ell_{p}$-spaces). Our results recover and in some case complete these old
results through a general approach on vector valued $m$-linear mappings.
