  Topological phases of matter is an exotic phenomena in modern condense matter
physics, which has attracted much attention due to the unique boundary states
and transport properties. Recently, this topological concept in electronic
materials has been exploited in many other fields of physics. Motivated by
designing and controlling the behavior of electromagnetic waves, in optical,
microwave, and sound frequencies, topological photonics emerges as a rapid
growing field of research. Due to the flexibility and diversity of
superconducting quantum circuits system, it is an promising platform to realize
exotic topological models and to probe and explore topologically-protected
effects in new ways. Here, we review theoretical and experimental advances of
topological photonics on superconducting quantum circuits via the
experimentally demonstrated parametric tunable coupling techniques, including
using of the superconducting transmission line resonator, superconducting
qubits, and the coupled system of them. On superconducting circuits, the
flexible interactions and intrinsic nonlinearity making topological photonics
in this system not only a simple photonic analog of topological effects for
novel devices, but also a realm of exotic but less-explored fundamental
physics.
