  Recent experimental advances in controlling dissipation have brought about
unprecedented flexibility in engineering non-Hermitian Hamiltonians in open
classical and quantum systems. A particular interest centers on the topological
properties of non-Hermitian systems, which exhibit unique phases with no
Hermitian counterparts. However, no systematic understanding in analogy with
the periodic table of topological insulators and superconductors has been
achieved. In this paper, we develop a coherent framework of topological phases
of non-Hermitian systems. After elucidating the physical meaning and the
mathematical definition of non-Hermitian topological phases, we start with
one-dimensional lattices, which exhibit topological phases with no Hermitian
counterparts and are found to be characterized by an integer topological
winding number even with no symmetry constraint, reminiscent of the quantum
Hall insulator in Hermitian systems. A system with a nonzero winding number,
which is experimentally measurable from the wave-packet dynamics, is shown to
be robust against disorder, a phenomenon observed in the Hatano-Nelson model
with asymmetric hopping amplitudes. We also unveil a novel bulk-edge
correspondence that features an infinite number of (quasi-)edge modes. We then
apply the K-theory to systematically classify all the non-Hermitian topological
phases in the Altland-Zirnbauer classes in all dimensions. The obtained
periodic table unifies time-reversal and particle-hole symmetries, leading to
highly nontrivial predictions such as the absence of non-Hermitian topological
phases in two dimensions. We provide concrete examples for all the nontrivial
non-Hermitian AZ classes in zero and one dimensions. In particular, we identify
a Z2 topological index for arbitrary quantum channels. Our work lays the
cornerstone for a unified understanding of the role of topology in
non-Hermitian systems.
