From author's abstract: We treat the question of the existence of solutions of boundary value problems for systems of nonlinear elliptic equations of the form \[ -\Delta u=f(x,u,v,\nabla u,\nabla v),\;-\Delta v=g(x,u,v,\nabla u,\nabla v),\;\text{in} \Omega . \] We discuss several classes of such systems using both variational and topological methods. The notion of criticality takes into consideration the coupling, which plays important roles in both a priori estimates for the solutions and Palais-Smale conditions for the associated functional in the variational case.