  We review recent developments concerning non-equilibrium quantum dynamics and
many-body physics with light, in superconducting circuits and Josephson
analogues. We start with quantum impurity models summarizing the effect of
dissipation and of driving the system. We mention theoretical and experimental
efforts to characterize these non-equilibrium quantum systems. We show how
Josephson junction systems can implement the equivalent of the Kondo effect
with microwave photons. The Kondo effect is characterized by a renormalized
light-frequency and a peak in the Rayleigh elastic transmission of a photon. We
also address the physics of hybrid systems comprising mesoscopic quantum dot
devices coupled to an electromagnetic resonator. Then, we discuss extensions to
Quantum Electrodynamics (QED) Networks allowing to engineer the Jaynes-Cummings
lattice and Rabi lattice models. This opens the door to novel many-body physics
with light out of equilibrium, in relation with the Mott-superfluid transition
observed with ultra-cold atoms in optical lattices. Then, we summarize recent
theoretical predictions for realizing topological phases with light. Synthetic
gauge fields and spin-orbit couplings have been successfully implemented with
ultra-cold atoms in optical lattices --- using time-dependent Floquet
perturbations periodic in time, for example --- as well as in photonic lattice
systems. Finally, we discuss the Josephson effect related to Bose-Hubbard
models in ladder and two-dimensional geometries. The Bose-Hubbard model is
related to the Jaynes-Cummings lattice model in the large detuning limit
between light and matter. In the presence of synthetic gauge fields, we show
that Meissner currents subsist in an insulating Mott phase.
