  We introduce non-linear versions of the classical cotype of Banach spaces. We
show that spaces with l.u.st and cotype, and that spaces having Fourier cotype
enjoy our non-linear cotype. We apply these concepts to get results on
convergence of vector-valued power series in infinite many variables and on
$\ell_{1}$-multipliers of vector-valued Dirichlet series. Finally we introduce
cotype with respect to indexing sets, an idea that includes our previous
definitions.
