  We consider fully nonlinear uniformly elliptic cooperative systems with
quadratic growth in the gradient, such as $$ -F_i(x, u_i, Du_i, D^2 u_i)-
\langle M_i(x)D u_i, D u_i \rangle =\lambda c_{i1}(x) u_1 + \cdots + \lambda
c_{in}(x) u_n +h_i(x), $$ for $i=1,\cdots,n$, in a bounded $C^{1,1}$ domain
$\Omega\subset \mathbb{R}^N$ with Dirichlet boundary conditions; here $n\geq
1$, $\lambda \in\mathbb{R}$, $c_{ij},\, h_i \in L^\infty(\Omega)$, $c_{ij}\geq
0$, $M_i$ satisfies $0<\mu_1 I\leq M_i\leq \mu_2 I$, and $F_i$ is an uniformly
elliptic Isaacs operator.
  We obtain uniform a priori bounds for systems, under a weak coupling
hypothesis that seems to be optimal. As an application, we also establish
existence and multiplicity results for these systems, including a branch of
solutions which is new even in the scalar case.
