  Active matter encompasses different nonequilibrium systems in which
individual constituents convert energy into non-conservative forces or motion
at the microscale. This review provides an elementary introduction to the role
of topology in active matter through experimentally relevant examples. Here,
the focus lies on topological defects and topologically protected edge modes
with an emphasis on the distinctive properties they acquire in active media.
These paradigmatic examples represent two physically distinct classes of
phenomena whose robustness can be traced to a common mathematical origin: the
presence of topological invariants. These invariants are typically integer
numbers that cannot be changed by continuous deformations of the relevant order
parameters or physical parameters of the underlying medium. We first explain
the mechanisms whereby topological defects self propel and proliferate in
active nematics, leading to collective states which can be manipulated by
geometry and patterning. Possible implications for active microfluidics and
biological tissues are presented. We then illustrate how the propagation of
waves in active fluids and solids is affected by the presence of topological
invariants characterizing their dispersion relations. We discuss the relevance
of these ideas for the design of robotic metamaterials and the properties of
active granular and colloidal systems. Open theoretical and experimental
challenges are presented as future research prospects.
