  We extend the notion of the Eigenstate Thermalization Hypothesis (ETH) to
Open Quantum Systems governed by the Gorini-Kossakowski-Lindblad-Sudarshan
(GKLS) Master Equation. We present evidence that the eigenstates of
non-equilibrium steady state (NESS) density matrices obey a generalization of
ETH in boundary-driven systems when the bulk Hamiltonian is non-integrable,
just as eigenstates of Gibbs density matrices are conjectured to do in
equilibrium. This generalized ETH, which we call NESS-ETH, can be used to
obtain representative pure states that reproduce the expectation values of
few-body operators in the NESS. The density matrices of these representative
pure states can be further interpreted as weak solutions of the GKLS Master
Equation. Additionally, we explore the validity and breakdown of NESS-ETH in
the presence of symmetries, integrability and many-body localization in the
bulk Hamiltonian.
