This review provides an overview of non-Hermitian classical and quantum physics, focusing on its foundations and applications. It begins with key mathematical concepts, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity, and parity-time (PT) symmetry. These concepts are essential for understanding non-Hermitian systems, which allow for non-conservation of probability and energy. The review discusses how diverse classical systems, such as photonics, mechanics, and acoustics, can be used to simulate non-Hermitian wave physics, highlighting phenomena like unidirectional invisibility and coherent perfect absorption. In quantum physics, non-Hermitian operators emerge as effective descriptions of open quantum systems, with applications in areas like quantum resonances and superradiance. The review also explores non-Hermitian quantum many-body physics, including quantum critical phenomena and nonunitary conformal field theories. A key focus is the concept of band topology in non-Hermitian systems, which extends to complex spectra and introduces new classifications and topological invariants. The review addresses various topics, including nonreciprocal transport, speed limits, and nonunitary quantum walks. It also discusses the implications of non-Hermitian physics for topological systems, such as edge modes and the bulk-edge correspondence. The review emphasizes the importance of non-Hermitian physics in bridging different branches of physics, from open quantum systems to classical optics and biophysics. It concludes with a summary of the field's current state and future directions, highlighting its potential for applications in quantum technologies and condensed matter physics.This review provides an overview of non-Hermitian classical and quantum physics, focusing on its foundations and applications. It begins with key mathematical concepts, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity, and parity-time (PT) symmetry. These concepts are essential for understanding non-Hermitian systems, which allow for non-conservation of probability and energy. The review discusses how diverse classical systems, such as photonics, mechanics, and acoustics, can be used to simulate non-Hermitian wave physics, highlighting phenomena like unidirectional invisibility and coherent perfect absorption. In quantum physics, non-Hermitian operators emerge as effective descriptions of open quantum systems, with applications in areas like quantum resonances and superradiance. The review also explores non-Hermitian quantum many-body physics, including quantum critical phenomena and nonunitary conformal field theories. A key focus is the concept of band topology in non-Hermitian systems, which extends to complex spectra and introduces new classifications and topological invariants. The review addresses various topics, including nonreciprocal transport, speed limits, and nonunitary quantum walks. It also discusses the implications of non-Hermitian physics for topological systems, such as edge modes and the bulk-edge correspondence. The review emphasizes the importance of non-Hermitian physics in bridging different branches of physics, from open quantum systems to classical optics and biophysics. It concludes with a summary of the field's current state and future directions, highlighting its potential for applications in quantum technologies and condensed matter physics.