  In this review we discuss the works in the area of quantum simulation and
many-body physics with light, from the early proposals on equilibrium models to
the more recent works in driven dissipative platforms. We start by describing
the founding works on Jaynes-Cummings-Hubbard model and the corresponding
photon-blockade induced Mott transitions and continue by discussing the
proposals to simulate effective spin models and fractional quantum Hall states
in coupled resonator arrays (CRAs). We also analyze the recent efforts to study
out-of-equilibrium many-body effects using driven CRAs, including the
predictions for photon fermionization and crystallization in driven rings of
CRAs as well as other dynamical and transient phenomena. We try to summarise
some of the relatively recent results predicting exotic phases such as
super-solidity and Majorana like modes and then shift our attention to
developments involving one dimensional nonlinear slow light setups. There the
simulation of strongly correlated phases characterising Tonks-Girardeau gases,
Luttinger liquids, and interacting relativistic fermionic models is described.
We review the major theory results and also briefly outline recent developments
in ongoing experimental efforts involving different platforms in circuit QED,
photonic crystals and nanophotonic fibers interfaced with cold atoms.
