  The Jaynes-Cummings model describes the coupling between photons and a single
two-level atom in a simplified representation of light-matter interactions. In
circuit QED, this model is implemented by combining microwave resonators and
superconducting qubits on a microchip with unprecedented experimental control.
Arranging qubits and resonators in the form of a lattice realizes a new kind of
Hubbard model, the Jaynes-Cummings-Hubbard model, in which the elementary
excitations are polariton quasi-particles. Due to the genuine openness of
photonic systems, circuit QED lattices offer the possibility to study the
intricate interplay of collective behavior, strong correlations and
non-equilibrium physics. Thus, turning circuit QED into an architecture for
quantum simulation, i.e., using a well-controlled system to mimic the intricate
quantum behavior of another system too daunting for a theorist to tackle
head-on, is an exciting idea which has served as theorists' playground for a
while and is now also starting to catch on in experiments. This review gives a
summary of the most recent theoretical proposals and experimental efforts in
this context.
