Waves are a universal manifestation of matter, ranging from surface waves on water and seismic waves to light and gravity, and from waves in media and structures to the microscopic behavior of particles. Unlike gravitational waves, mechanical waves depend on the properties of the medium and the structure through which they propagate. It is therefore not surprising that the diversity of media leads to a wide variety of mechanical wave types. This book presents a comprehensive and unified framework for understanding mechanical wave phenomena. Covering both classical foundations and advanced developments, it systematically addresses general wave properties and the specific characteristics of waves in a wide range of media and structures—both continuous and discrete. The text is organized into eight parts, beginning with a concise introduction, On Waves in Brief, which highlights key concepts such as wave types, phase and group velocities, and wave propagation in different media. As the first level of wave theory, this introductory section is designed to be accessible to a broad readership, while the subsequent chapters examine specialized topics intended for researchers, engineers, and graduate students. Core themes include wave energy, mass, and momentum, with particular attention to novel topics such as wave energy partitioning, the modern formulation of wave momentum, and its implications for wave dynamics. The book also explores wave dispersion, orthogonality and localization, general dynamic elasticity, and nonlinear effects in mechanical systems. In addition, the treatment of forced waves provides new insights into excitation mechanisms, phase shifts, resonance phenomena, and the manifestation of the causality principle. By integrating classical results with recent advances—some presented here for the first time—this book offers a coherent theoretical framework that connects mechanical waves with broader wave phenomena, from gravitational and electromagnetic waves to localized vibrations in structured materials. It serves as a valuable reference for researchers in wave mechanics, materials science, and applied physics, as well as for graduate students seeking a deeper understanding of mechanical wave theory.