  The synergy between non-Hermitian concepts and topological ideas have led to
very fruitful activity in the recent years. Their interplay has resulted in a
wide variety of new non-Hermitian topological phenomena being discovered. In
this review, we present the key principles underpinning the topological
features of non-Hermitian phases. Using paradigmatic models -- Hatano-Helson,
non-Hermitian Su-Schrieffer-Heeger and non-Hermitian Chern insulator -- we
illustrate the central features of non-Hermitian topological systems, including
exceptional points, complex energy gaps and non-Hermitian symmetry
classification. We discuss the non-Hermitian skin effect and the notion of the
generalized Brillouin zone, which allows restoring the bulk-boundary
correspondence. Using concrete examples, we examine the role of disorder,
present the linear response framework, and analyze the Hall transport
properties of non-Hermitian topological systems. We also survey the rapidly
growing experimental advances in this field. Finally, we end by highlighting
possible directions which, in our view, may be promising for explorations in
the near future.
