This review provides a comprehensive overview of non-Hermitian physics, covering both classical and quantum systems. It begins by introducing key theorems and concepts in non-Hermitian linear algebra, such as Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity, and parity-time symmetry. These concepts are then applied to various classical systems, including photonics, mechanics, electrical circuits, acoustics, and active matter, highlighting unique phenomena like unidirectional invisibility, enhanced sensitivity, topological energy transfer, coherent perfect absorption, single-mode lasing, and robust biological transport. The review also delves into the quantum regime, explaining how non-Hermitian operators emerge as effective descriptions of open quantum systems through the Feshbach projection approach and the quantum trajectory approach. It discusses their applications in atomic, molecular, and optical physics, mesoscopic physics, and nuclear physics, focusing on phenomena such as quantum resonances, superradiance, continuous quantum Zeno effect, quantum critical phenomena, Dirac spectra in quantum chromodynamics, and nonunitary conformal field theories. A significant portion of the review is dedicated to the concept of band topology in complex spectra of non-Hermitian systems, presenting classifications and examples. The bulk-edge correspondence and topological invariants are discussed, along with the proof of these results. Other topics covered include nonreciprocal transport, speed limits, nonunitary quantum walk, and miscellaneous subjects related to non-Hermitian physics. The review aims to bridge the gap between different branches of physics, providing a coherent and self-contained guide to the field of non-Hermitian physics. It includes detailed mathematical foundations, practical applications, and a broad range of examples to illustrate the concepts and phenomena discussed.This review provides a comprehensive overview of non-Hermitian physics, covering both classical and quantum systems. It begins by introducing key theorems and concepts in non-Hermitian linear algebra, such as Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity, and parity-time symmetry. These concepts are then applied to various classical systems, including photonics, mechanics, electrical circuits, acoustics, and active matter, highlighting unique phenomena like unidirectional invisibility, enhanced sensitivity, topological energy transfer, coherent perfect absorption, single-mode lasing, and robust biological transport. The review also delves into the quantum regime, explaining how non-Hermitian operators emerge as effective descriptions of open quantum systems through the Feshbach projection approach and the quantum trajectory approach. It discusses their applications in atomic, molecular, and optical physics, mesoscopic physics, and nuclear physics, focusing on phenomena such as quantum resonances, superradiance, continuous quantum Zeno effect, quantum critical phenomena, Dirac spectra in quantum chromodynamics, and nonunitary conformal field theories. A significant portion of the review is dedicated to the concept of band topology in complex spectra of non-Hermitian systems, presenting classifications and examples. The bulk-edge correspondence and topological invariants are discussed, along with the proof of these results. Other topics covered include nonreciprocal transport, speed limits, nonunitary quantum walk, and miscellaneous subjects related to non-Hermitian physics. The review aims to bridge the gap between different branches of physics, providing a coherent and self-contained guide to the field of non-Hermitian physics. It includes detailed mathematical foundations, practical applications, and a broad range of examples to illustrate the concepts and phenomena discussed.