This review article, authored by Kostas D. Kokkotas and Bernd G. Schmidt, provides a comprehensive overview of the theory of quasi-normal modes (QNMs) of compact objects, including both black holes and relativistic stars. QNMs are oscillations that occur in the spacetime around these objects due to gravitational wave emissions, and they are of significant interest in the context of gravitational wave astronomy. The article covers the mathematical and astrophysical aspects of QNMs, discussing the properties of various families of QNMs and numerical techniques for calculating them. It also reviews the successes and limitations of perturbation theory, and its role in the emerging era of numerical relativity and supercomputers. The authors delve into the perturbations of different types of black holes (Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman) and relativistic stars (non-rotating and slowly-rotating), providing detailed analyses of their QNMs. Additionally, the article explores the excitation and detection of QNMs, as well as the numerical techniques used in their study. The review concludes with a discussion on the future directions, including the synergism between perturbation theory and numerical relativity, second-order perturbations, mode calculations, and the role of detectors in gravitational wave astronomy.This review article, authored by Kostas D. Kokkotas and Bernd G. Schmidt, provides a comprehensive overview of the theory of quasi-normal modes (QNMs) of compact objects, including both black holes and relativistic stars. QNMs are oscillations that occur in the spacetime around these objects due to gravitational wave emissions, and they are of significant interest in the context of gravitational wave astronomy. The article covers the mathematical and astrophysical aspects of QNMs, discussing the properties of various families of QNMs and numerical techniques for calculating them. It also reviews the successes and limitations of perturbation theory, and its role in the emerging era of numerical relativity and supercomputers. The authors delve into the perturbations of different types of black holes (Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman) and relativistic stars (non-rotating and slowly-rotating), providing detailed analyses of their QNMs. Additionally, the article explores the excitation and detection of QNMs, as well as the numerical techniques used in their study. The review concludes with a discussion on the future directions, including the synergism between perturbation theory and numerical relativity, second-order perturbations, mode calculations, and the role of detectors in gravitational wave astronomy.