This review presents a comprehensive theoretical description of the many-body dynamical electronic response of solids, focusing on collective electronic excitations at metal surfaces, including the conventional surface plasmon, multipole plasmons, and the recently predicted acoustic surface plasmon. The paper discusses both theoretical calculations and experimental measurements, as well as their applications in various fields such as electrochemistry, biosensing, and plasmonics. It also reviews the classical and nonlocal models for surface plasmons, including the surface-plasmon polariton, and explores the dispersion relations, energy loss, and coupling mechanisms of these excitations. The paper highlights the importance of surface plasmons in understanding various physical phenomena, such as Van der Waals forces, energy transfer in gas-surface interactions, and surface energies. It also discusses the role of surface plasmons in modern technologies, including plasmonics, which enables the manipulation of light at the nanoscale. The review emphasizes the significance of surface plasmons in both fundamental research and practical applications, and provides a detailed analysis of their properties, including their dispersion, damping, and interaction with external fields. The paper also addresses the challenges in understanding and modeling surface plasmons, particularly in complex systems such as thin films and nanostructured materials. Overall, the review provides a thorough overview of the current state of knowledge on surface plasmons and their potential for future research and applications.This review presents a comprehensive theoretical description of the many-body dynamical electronic response of solids, focusing on collective electronic excitations at metal surfaces, including the conventional surface plasmon, multipole plasmons, and the recently predicted acoustic surface plasmon. The paper discusses both theoretical calculations and experimental measurements, as well as their applications in various fields such as electrochemistry, biosensing, and plasmonics. It also reviews the classical and nonlocal models for surface plasmons, including the surface-plasmon polariton, and explores the dispersion relations, energy loss, and coupling mechanisms of these excitations. The paper highlights the importance of surface plasmons in understanding various physical phenomena, such as Van der Waals forces, energy transfer in gas-surface interactions, and surface energies. It also discusses the role of surface plasmons in modern technologies, including plasmonics, which enables the manipulation of light at the nanoscale. The review emphasizes the significance of surface plasmons in both fundamental research and practical applications, and provides a detailed analysis of their properties, including their dispersion, damping, and interaction with external fields. The paper also addresses the challenges in understanding and modeling surface plasmons, particularly in complex systems such as thin films and nanostructured materials. Overall, the review provides a thorough overview of the current state of knowledge on surface plasmons and their potential for future research and applications.