  This is an introductory review of the physics of topological quantum matter
with cold atoms. Topological quantum phases, originally discovered and
investigated in condensed matter physics, have recently been explored in a
range of different systems, which produced both fascinating physics findings
and exciting opportunities for applications. Among the physical systems that
have been considered to realize and probe these intriguing phases, ultracold
atoms become promising platforms due to their high flexibility and
controllability. Quantum simulation of topological phases with cold atomic
gases is a rapidly evolving field, and recent theoretical and experimental
developments reveal that some toy models originally proposed in condensed
matter physics have been realized with this artificial quantum system. The
purpose of this article is to introduce these developments. The article begins
with a tutorial review of topological invariants and the methods to control
parameters in the Hamiltonians of neutral atoms. Next, topological quantum
phases in optical lattices are introduced in some detail, especially several
celebrated models, such as the Su-Schrieffer-Heeger model, the
Hofstadter-Harper model, the Haldane model and the Kane-Mele model. The
theoretical proposals and experimental implementations of these models are
discussed. Notably, many of these models cannot be directly realized in
conventional solid-state experiments. The newly developed methods for probing
the intrinsic properties of the topological phases in cold atom systems are
also reviewed. Finally, some topological phases with cold atoms in the
continuum and in the presence of interactions are discussed, and an outlook on
future work is given.
