All real materials in one way or another, exhibit a departure from ideal elastic behaviour, even at very small strain values. Under cyclic deformation, these departtures result in irreversible energy losses in material. The causes of such losses are many, and include the irreversible transfer of mechanical energy into heat, growth of cracks and other defects, and the microplastic deformaton of crystals to name a few. Several terms have been suggested to define these phenomena including damping, energy dissipation, imperfect elasticity and internal friction. This book is about materials damping; with damping or energy dissipation processes in vibrating solids.