We study nonlinear systems of the form $-\Delta_pu=v^{q_1}+\mu,\; -\Delta_pv=u^{q_2}+\eta$ and $F_k[-u]=v^{s_1}+\mu,\; F_k[-v]=u^{s_2}+\eta$ in a bounded domain $\Omega$ or in $\mathbb{R}^N$ where $\mu$ and $\eta$ are nonnegative Radon measures, $\Delta_p$ and $F_k$ are respectively the $p$-Laplacian and the $k$-Hessian operators and $q_1$, $q_2$, $s_1$ and $s_2$ positive numbers. We give necessary and sufficient conditions for existence expressed in terms of Riesz or Bessel capacities.