  Self-propulsion is a quintessential aspect of biological systems, which can
induce nonequilibrium phenomena that have no counterparts in passive systems.
Motivated by biophysical interest together with recent advances in experimental
techniques, active matter has been a rapidly developing field in physics.
Meanwhile, over the past few decades, topology has played a crucial role to
understand certain robust properties appearing in condensed matter systems. For
instance, the nontrivial topology of band structures leads to the notion of
topological insulators, where one can find robust gapless edge modes protected
by the bulk band topology. We here review recent progress in an
interdisciplinary area of research at the intersection of these two fields.
Specifically, we give brief introductions to active matter and band topology in
Hermitian systems, and then explain how the notion of band topology can be
extended to nonequilibrium (and thus non-Hermitian) systems including active
matter. We review recent studies that have demonstrated the intimate
connections between active matter and topological materials, where exotic
topological phenomena that are unfeasible in passive systems have been found. A
possible extension of the band topology to nonlinear systems is also briefly
discussed. Active matter can thus provide an ideal playground to explore
topological phenomena in qualitatively new realms beyond conservative linear
systems.
