Exceptional points (EPs) are singularities in eigenvalue problems where eigenvalues and eigenvectors coalesce. They occur in various physical systems, including classical and quantum mechanics, and are associated with symmetry breaking in PT-symmetric Hamiltonians. EPs are linked to quantum phase transitions, quantum chaos, and dramatic effects in multichannel scattering. In optics, EPs are observed in systems with gain and loss, leading to phenomena like resonance narrowing and laser oscillation. In atomic and molecular physics, EPs influence processes such as resonant dissociation and Bose-Einstein condensate dynamics. In nuclear physics, EPs are related to instabilities and continuum problems. EPs also affect approximation schemes and have implications for the convergence of perturbation series. The physical manifestation of EPs has been demonstrated in experiments, particularly in microwave cavities and optical systems. EPs are characterized by square root branch points and have significant effects on observables, including the behavior of Green's functions and scattering matrices. In quantum systems, EPs are crucial for understanding phase transitions and chaos. The study of EPs has led to new insights in both theoretical and experimental physics, with applications in quantum mechanics, optics, and other fields.Exceptional points (EPs) are singularities in eigenvalue problems where eigenvalues and eigenvectors coalesce. They occur in various physical systems, including classical and quantum mechanics, and are associated with symmetry breaking in PT-symmetric Hamiltonians. EPs are linked to quantum phase transitions, quantum chaos, and dramatic effects in multichannel scattering. In optics, EPs are observed in systems with gain and loss, leading to phenomena like resonance narrowing and laser oscillation. In atomic and molecular physics, EPs influence processes such as resonant dissociation and Bose-Einstein condensate dynamics. In nuclear physics, EPs are related to instabilities and continuum problems. EPs also affect approximation schemes and have implications for the convergence of perturbation series. The physical manifestation of EPs has been demonstrated in experiments, particularly in microwave cavities and optical systems. EPs are characterized by square root branch points and have significant effects on observables, including the behavior of Green's functions and scattering matrices. In quantum systems, EPs are crucial for understanding phase transitions and chaos. The study of EPs has led to new insights in both theoretical and experimental physics, with applications in quantum mechanics, optics, and other fields.