We develop an exact non-perturbative framework to compute steady state properties of quantum-impurities connected to leads subject to finite bias. We show that the steady-state physics of these systems is captured by nonequilibrium scattering eigenstates defined on the infinite open line with boundary conditions set by the leads. Introducing the Scattering Bethe Ansatz (SBA), a generalization to the open system of the traditional Bethe-Ansatz, we show how to explicitly construct the fully interacting, current-carrying and entropy producing scattering eigenstates. As an example we derive exact results for the steady state properties of the Interacting Resonance Level model.