This is an applied mathematics project aimed at the application of advanced kinetic theory for semiconductor modelling. The analytical tools are partial differential equations in a functional analytic framework and semiclassical limits in the Wigner formulation of quantum physics. The main objectives of this TMR proposal are a) training of the proposer in the analytical and numerical methods pursued at the host group, b) obtaining new results in the mathematical analysis of quantum kinetic equations, c) deriving improved models and numerical methods for the simulation of state-of-the art semiconductor devices and bridging the gap between rigorous mathematics and approaches pursued by physicists and engineers, d) fostering the European cooperation in Applied Mathematics. The home group of the proposer focuses on the analysis of quantum kinetic and quantum hydrodynamic equations. The host group is internationally well known for work on semiconductor Boltzmann equations and related models, including significant contributions in their numerical analysis.