Quasi-normal modes (QNMs) of stars and black holes are oscillations that occur in compact objects due to perturbations. These modes are complex frequencies, with the real part representing the oscillation frequency and the imaginary part representing the damping. QNMs are important in gravitational wave astronomy as they provide a unique signature of the object's properties. This review discusses the theory of QNMs for both black holes and relativistic stars, covering their properties, calculation methods, and relevance to gravitational wave detection. The discussion includes black holes such as Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman, as well as relativistic stars. The review also addresses the stability and completeness of QNMs, the excitation and detection of QNMs, and the numerical techniques used to calculate them. The review highlights the successes and limitations of perturbation theory in the context of QNMs and their role in the emerging era of numerical relativity and supercomputers. The study of QNMs has been crucial in understanding the behavior of compact objects and their interactions with gravitational waves. The review also discusses the mathematical and physical significance of QNMs, their relation to resonances, and their importance in astrophysics. The review concludes with a discussion of the future directions in the study of QNMs and their potential applications in gravitational wave astronomy.Quasi-normal modes (QNMs) of stars and black holes are oscillations that occur in compact objects due to perturbations. These modes are complex frequencies, with the real part representing the oscillation frequency and the imaginary part representing the damping. QNMs are important in gravitational wave astronomy as they provide a unique signature of the object's properties. This review discusses the theory of QNMs for both black holes and relativistic stars, covering their properties, calculation methods, and relevance to gravitational wave detection. The discussion includes black holes such as Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman, as well as relativistic stars. The review also addresses the stability and completeness of QNMs, the excitation and detection of QNMs, and the numerical techniques used to calculate them. The review highlights the successes and limitations of perturbation theory in the context of QNMs and their role in the emerging era of numerical relativity and supercomputers. The study of QNMs has been crucial in understanding the behavior of compact objects and their interactions with gravitational waves. The review also discusses the mathematical and physical significance of QNMs, their relation to resonances, and their importance in astrophysics. The review concludes with a discussion of the future directions in the study of QNMs and their potential applications in gravitational wave astronomy.