Numerical simulations are becoming an indispensable tool in many applications involving processing, manufacturing, and performance of metallic and composite materials. The corresponding tools are based on a number of fundamental relationships that were extracted from mechanics (and underlying physics). These relationships are universal with respect to materials but they are not adequate to solve boundary value problems in which constitutive descriptions of deformation and failure are essential. The constitutive descriptions must not only account for the physical mechanisms but also for the input data needed for identification of the material coefficients. Thus, to accomplish this, the relationship between the variables and their rates must comply with the materials response within the framework set by mechanics and thermodynamic principles. A purely numerical description, which relates the state of a material to a set of data, is likely to be empirical. It leads to unreliable extrapolation of states, which are not included in the database. Other considerations on constitutive models are the scale of the material features needed, the size of the structure and the associated computation time. The scale is dictated by the smallest microstructural information needed to characterize a product e.g., the grain size, which then enters simulations at the meso-scale. It is possible to envision cases were details at a finer scale are necessary, for instance for the manufacturing of micro-devices. Even if the idea of defining the constitutive description from the atomic scale is scientifically and philosophically very attractive, it is usually not practical. The course will focus on descriptions of critical states for advanced metallic materials and composites. The approaches taken for this purpose can be categorized as follows: strongly based on Continuum Mechanics, incorporating knowledge of microstructure, and applying homogenization and other numerical approaches. The course will introduce the classical approaches and treat the new developments in a critical manner. Obviously, application of advanced materials rests on efficient and physically based constitutive relations. The following specific topics will be covered: Plastic behavior at non-proportional loading incorporating the influence of relevant microstructural features, numerical simulation of metal forming of advanced high strength steels, termination of elastic range of pressure insensitive and sensitive materials, anisotropic vs. isotropic initial yield/failure criteria, structural components subjected to high temperatures, mechanical and thermal cyclic loads on the components under creep conditions, phase mixture model for simulating the mechanical behavior of tempered martensitic steels at high temperatures, but moderate mechanical loads, discretization methods for elasto-plastic solids, integration of plasticity models for finite loading steps, volumetric locking for fully developed plastic flow, mechanisms based modelling of failure in composite materials, basic aspects of fracture mechanics, and fracture and damage criteria. The course is addressed to Master Course students of Mechanical and Civil Engineering as well as Computational Mechanics, PhD-students, young scientists, and research engineers.