  The current understanding of the role of topology in non-Hermitian (NH)
systems and its far-reaching physical consequences observable in a range of
dissipative settings are reviewed. In particular, how the paramount and
genuinely NH concept of exceptional degeneracies, at which both eigenvalues and
eigenvectors coalesce, leads to phenomena drastically distinct from the
familiar Hermitian realm is discussed. An immediate consequence is the
ubiquitous occurrence of nodal NH topological phases with concomitant open
Fermi-Seifert surfaces, where conventional band-touching points are replaced by
the aforementioned exceptional degeneracies. Furthermore, new notions of gapped
phases including topological phases in single-band systems are detailed, and
the manner in which a given physical context may affect the symmetry-based
topological classification is clarified. A unique property of NH systems with
relevance beyond the field of topological phases consists of the anomalous
relation between bulk and boundary physics, stemming from the striking
sensitivity of NH matrices to boundary conditions. Unifying several
complementary insights recently reported in this context, a picture of
intriguing phenomena such as the NH bulk-boundary correspondence and the NH
skin effect is put together. Finally, applications of NH topology in both
classical systems including optical setups with gain and loss, electric
circuits,s and mechanical systems and genuine quantum systems such as
electronic transport settings at material junctions and dissipative cold-atom
setups are reviewed.
