In this talk, I will present recent results on Liouville-type theorems for the steady incompressible Navier-Stokes system in a three-dimensional slab with either no-slip boundary conditions or periodic boundary conditions. When the no-slip boundary conditions are prescribed, we prove that any bounded solution is trivial if it is axisymmetric or $ru^r$ is bounded, and that general three-dimensional solutions must be Poiseuille flows when the velocity is not big. When the periodic boundary conditions are imposed on the slab boundaries, we prove that the bounded solutions must be constant vectors if either the swirl velocity is independent of the angular variable, or $ru^r$ decays to zero as $r$ tends to infinity, The proofs are based on the fundamental structure of the equations and energy estimates. The key technique is to establish a Saint-Venant type estimate that characterizes the growth of Dirichlet integral of nontrivial solutions. The talk is based on a recent joint work with Jeaheang Bang, Yun Wang and Chunjing Xie.