  We develop a complete theory of symmetry and topology in non-Hermitian
physics. We demonstrate that non-Hermiticity ramifies the celebrated
Altland-Zirnbauer symmetry classification for insulators and superconductors.
In particular, charge conjugation is defined in terms of transposition rather
than complex conjugation due to the lack of Hermiticity, and hence chiral
symmetry becomes distinct from sublattice symmetry. It is also shown that
non-Hermiticity enables a Hermitian-conjugate counterpart of the
Altland-Zirnbauer symmetry. Taking into account sublattice symmetry or
pseudo-Hermiticity as an additional symmetry, the total number of symmetry
classes is 38 instead of 10, which describe intrinsic non-Hermitian topological
phases as well as non-Hermitian random matrices. Furthermore, due to the
complex nature of energy spectra, non-Hermitian systems feature two different
types of complex-energy gaps, point-like and line-like vacant regions. On the
basis of these concepts and K-theory, we complete classification of
non-Hermitian topological phases in arbitrary dimensions and symmetry classes.
Remarkably, non-Hermitian topology depends on the type of complex-energy gaps
and multiple topological structures appear for each symmetry class and each
spatial dimension, which are also illustrated in detail with concrete examples.
Moreover, the bulk-boundary correspondence in non-Hermitian systems is
elucidated within our framework, and symmetries preventing the non-Hermitian
skin effect are identified. Our classification not only categorizes recently
observed lasing and transport topological phenomena, but also predicts a new
type of symmetry-protected topological lasers with lasing helical edge states
and dissipative topological superconductors with nonorthogonal Majorana edge
states. Furthermore, our theory provides topological classification of
Hermitian and non-Hermitian free bosons.
