We develop an exact non-perturbative framework to compute the nonequilibrium steady state properties of quantum impurities connected to leads subject to source-drain voltage. We show that in the open system limit the non-equilibrium physics is captured by eigenstates defined with boundary conditions set by the leads. The eigenstates are current carrying and entropy producing, with the dissipation inherent in the limit. We construct these eigenstates by means of a recently introduced Scattering (or Open) Bethe Ansatz approach, a generalization to nonequilibrium of the Thermodynamic (or Closed) Bethe Ansatz. We compute the I(V) curve of the Interacting Resonance Level and observe a Fermi Edge Singularity out of equilibrium as the impurity level approaches resonance. We then apply the approach to the quantum dot (the nonequilibrium Anderson Impurity model) and compute the conductance G(V ; e_d), noting the formation of the Kondo peak as the gate voltage e_d is decreased, and the peak destruction as the source-drain voltage V is increased.