This review provides an introduction to quasinormal modes (QNMs) of black holes and black branes, focusing on recent developments in the field. QNMs are eigenmodes of dissipative systems, naturally arising from perturbations of classical gravitational backgrounds involving black holes or branes. They are crucial for understanding the near-equilibrium properties of strongly coupled quantum field theories and have significant implications in astrophysics, particularly in gravitational wave astronomy and the gauge-gravity duality. The review covers the following key areas: 1. **Perturbation Theory Primer**: It discusses the methods used to analyze perturbations of black hole backgrounds, including the Schwarzschild-anti-de Sitter (SAdS) geometry, higher-dimensional gravitational perturbations, and weak fields in the Kerr background. 2. **Definition of QNMs**: It explains how QNMs are defined as eigenmodes of non-Hermitian differential equations and as poles in the Green's function. 3. **Computing QNMs**: The methods for computing QNMs, including exact solutions, the WKB approximation, monodromy techniques, and series solutions, are covered. 4. **Spectra of Black Holes**: The review details the QNM spectra for various black hole backgrounds, such as Schwarzschild, Kerr, and asymptotically AdS black holes. 5. **Gauge-Gravity Duality**: It explores the connection between QNMs and the AdS/CFT correspondence, highlighting universality in the gravitational frequency of black branes and the role of QNMs in understanding hydrodynamics and the no-hair theorem. 6. **Astrophysical Applications**: The review discusses how QNMs can be used to infer the mass and spin of astrophysical black holes, test general relativity, and detect black holes through gravitational waves. 7. **Recent Developments**: It covers recent advancements in the field, including the role of highly damped QNMs, connections to phase transitions, and applications in analogue gravity and higher-dimensional spacetimes. The review aims to provide a comprehensive overview of the current state of QNM research, making it accessible to both experts and newcomers to the field.This review provides an introduction to quasinormal modes (QNMs) of black holes and black branes, focusing on recent developments in the field. QNMs are eigenmodes of dissipative systems, naturally arising from perturbations of classical gravitational backgrounds involving black holes or branes. They are crucial for understanding the near-equilibrium properties of strongly coupled quantum field theories and have significant implications in astrophysics, particularly in gravitational wave astronomy and the gauge-gravity duality. The review covers the following key areas: 1. **Perturbation Theory Primer**: It discusses the methods used to analyze perturbations of black hole backgrounds, including the Schwarzschild-anti-de Sitter (SAdS) geometry, higher-dimensional gravitational perturbations, and weak fields in the Kerr background. 2. **Definition of QNMs**: It explains how QNMs are defined as eigenmodes of non-Hermitian differential equations and as poles in the Green's function. 3. **Computing QNMs**: The methods for computing QNMs, including exact solutions, the WKB approximation, monodromy techniques, and series solutions, are covered. 4. **Spectra of Black Holes**: The review details the QNM spectra for various black hole backgrounds, such as Schwarzschild, Kerr, and asymptotically AdS black holes. 5. **Gauge-Gravity Duality**: It explores the connection between QNMs and the AdS/CFT correspondence, highlighting universality in the gravitational frequency of black branes and the role of QNMs in understanding hydrodynamics and the no-hair theorem. 6. **Astrophysical Applications**: The review discusses how QNMs can be used to infer the mass and spin of astrophysical black holes, test general relativity, and detect black holes through gravitational waves. 7. **Recent Developments**: It covers recent advancements in the field, including the role of highly damped QNMs, connections to phase transitions, and applications in analogue gravity and higher-dimensional spacetimes. The review aims to provide a comprehensive overview of the current state of QNM research, making it accessible to both experts and newcomers to the field.