  A review is given on the foundations and applications of non-Hermitian
classical and quantum physics. First, key theorems and central concepts in
non-Hermitian linear algebra, including Jordan normal form, biorthogonality,
exceptional points, pseudo-Hermiticity and parity-time symmetry, are delineated
in a pedagogical and mathematically coherent manner. Building on these, we
provide an overview of how diverse classical systems, ranging from photonics,
mechanics, electrical circuits, acoustics to active matter, can be used to
simulate non-Hermitian wave physics. In particular, we discuss rich and unique
phenomena found therein, such as unidirectional invisibility, enhanced
sensitivity, topological energy transfer, coherent perfect absorption,
single-mode lasing, and robust biological transport. We then explain in detail
how non-Hermitian operators emerge as an effective description of open quantum
systems on the basis of the Feshbach projection approach and the quantum
trajectory approach. We discuss their applications to physical systems relevant
to a variety of fields, including atomic, molecular and optical physics,
mesoscopic physics, and nuclear physics with emphasis on prominent
phenomena/subjects in quantum regimes, such as quantum resonances,
superradiance, continuous quantum Zeno effect, quantum critical phenomena,
Dirac spectra in quantum chromodynamics, and nonunitary conformal field
theories. Finally, we introduce the notion of band topology in complex spectra
of non-Hermitian systems and present their classifications by providing the
proof, firstly given by this review in a complete manner, as well as a number
of instructive examples. Other topics related to non-Hermitian physics,
including nonreciprocal transport, speed limits, nonunitary quantum walk, are
also reviewed.
