The monograph presents the results of research and development to create a theory of damage and destruction of deformable isotropic and anisotropic materials. Models and criteria are described that allow predicting destruction at monotonous, nonmonotonic, including sign-variable, cyclic load. From the proposed criteria in extreme particular cases follow the well-known criteria used in the prediction of destruction at monotonous, as well as cyclic stress, leading to fatigue destruction. The possibility of applying the proposed models and criteria for the design and optimization of materials processing processes is shown . For students, undergraduates, graduate students, engineers and researchers.