  Discrete quantum walks are dynamical protocols for controlling a single
quantum particle. Despite of its simplicity, quantum walks display rich
topological phenomena and provide one of the simplest systems to study and
understand topological phases. In this article, we review the physics of
discrete quantum walks in one and two dimensions in light of topological
phenomena and provide elementary explanations of topological phases and their
physical consequence, namely the existence of boundary states. We demonstrate
that quantum walks are versatile systems that simulate many topological phases
whose classifications are known for static Hamiltonians. Furthermore,
topological phenomena appearing in quantum walks go beyond what has been known
in static systems; there are phenomena unique to quantum walks, being an
example of periodically driven systems, that do not exist in static systems.
Thus the quantum walks not only provide a powerful tool as a quantum simulator
for static topological phases but also give unique opportunity to study
topological phenomena in driven systems.
