This volume presents asymptotic homogenization methods for the analysis of wave propagation in metamaterials and structured media. It introduces the foundations of two-scale asymptotic approaches and shows how periodic microstructures can be replaced by effective models that retain the essential physical mechanisms while substantially reducing the complexity of numerical simulations. Particular attention is devoted to both volume homogenization, leading to effective propagation equations, and interface homogenization, in which thin structured layers are represented by effective transmission or jump conditions. The book also examines configurations involving combined bulk and boundary effects, including finite-thickness structures and critical scattering regimes. The methods are illustrated through applications in acoustics, elastodynamics, and electromagnetism, covering both non-resonant and resonant metamaterials. The topics addressed include spoof plasmons supported by structured boundaries, localized surface plasmon resonances, elastic plate arrays on soils, acoustic labyrinthine metamaterials and metacrystals, and subwavelength arrays of Helmholtz resonators. Throughout the volume, detailed derivations are provided in order to make recent homogenization techniques accessible beyond a specialist readership. By connecting microscopic descriptions with effective macroscopic models, the book offers a unified framework for understanding and predicting wave phenomena in a broad range of structured media.