The purpose of the project is to study linear and nonlinear models arising in quantum mechanics and which are used to describematter at the microscopic and nanoscopic scales. The project focuses on physically-oriented questions (rigorous derivation of agiven model from first principles), analytic problems (existence and properties of bound states, study of solutions to timedependentequations) and numerical issues (development of reliable algorithmic strategies). Most of the models are nonlinear anddescribe physical systems possessing an infinite number of quantum particles, leading to specific difficulties.The first part of the project is devoted to the study of relativistic atoms and molecules, while taking into account quantumelectrodynamics effects like the polarization of the vacuum. The models are all based on the Dirac operator.The second part is focused on the study of quantum crystals. The goal is to develop new strategies for describing their behavior inthe presence of defects and local deformations. Both insulators, semiconductors and metals are considered (including graphene).In the third part, attractive systems are considered (like stars or a few nucleons interacting via strong forces in a nucleus). Theproject aims at rigorously understanding some of their specific properties, like Cooper pairing or the possible dynamical collapse ofmassive gravitational objects.Finally, the last part is devoted to general properties of infinite quantum systems, in particular the proof of the existence of thethermodynamic limit