  Thouless pumping provides one of the simplest manifestations of topology in
quantum systems, and has attracted a lot of recent interest, both theoretically
and experimentally. Since the seminal works by Thouless and Niu in 1983 and
1984, it is argued that the quantization of the pumped charge is robust against
weak disorder, but a clear characterization of the localization properties of
the relevant states, and the breakdown of quantized transport in the presence
of interaction or out of the adiabatic approximation, has been long debated.
Thouless pumping is also the first example of a topological phase emerging in a
periodically-driven system. Driven systems can exhibit exotic topological
phases without any static analogue and have been the subject of many recent
proposals both in fermionic and bosonic systems in diverse platforms ranging
from cold atoms to photonics and condensed matter systems. In this respect,
this review has a twofold purpose: On the one hand, it serves as a basis to
understand the robustness of the topology of slowly-driven systems per se; On
the other hand, it highlights the rich properties of topological pumps and
their diverse range of applications, for instance, in systems with synthetic
dimensions or for understanding higher-order topological phases. These examples
underline the relevance of topological pumping for the fast growing field of
topological quantum matter.
