  Quantum simulators are attractive as a means to study many-body quantum
systems that are not amenable to classical numerical treatment. A versatile
framework for quantum simulation is offered by superconducting circuits. In
this perspective, we discuss how superconducting circuits allow the engineering
of a wide variety of interactions, which in turn allows the simulation of a
wide variety of model Hamiltonians. In particular we focus on strong
photon-photon interactions mediated by nonlinear elements. This includes
on-site, nearest-neighbour and four-body interactions in lattice models,
allowing the implementation of extended Bose-Hubbard models and the toric code.
We discuss not only the present state in analogue quantum simulation, but also
future perspectives of superconducting quantum simulation that open up when
concatenating quantum gates in emerging quantum computing platforms.
