  We recapitulate recent developments of the functional renormalization group
(FRG) approach to the steady state of systems out of thermal equilibrium. In
particular, we discuss second-order truncation schemes which account for the
frequency-dependence of the two particle vertex and which incorporate inelastic
processes. Our focus is on two different types of one-dimensional fermion
chains: i) infinite, open systems which feature a translation symmetry, and ii)
finite systems coupled to left and right reservoirs. In addition to giving a
detailed and unified review of the technical derivation of the FRG schemes, we
briefly summarize some of the key physical results. In particular, we compute
the non-equilibrium phase diagram and analyze the fate of the
Berezinskii-Kosterlitz-Thouless transition in the infinite, open system.
