Mathematical concepts of quantum mechanics The book gives a streamlined introduction to quantum mechanics while describing the basic mathematical structures underpinning this discipline. Starting with an overview of key physical experiments illustrating the origin of the physical foundations the book proceeds with a description of the basic notions of quantum mechanics and their mathematical content. It then makes its way to topics of current interest specifically those in which mathematics plays an important role. The more advanced topics presented include: many body systems modern perturbation theory path integrals the theory of resonances adiabatic theory geometrical phases Aharonov Bohm effect density functional theory open systems the theory of radiation (non relativistic quantum electrodynamics) and the renormalization group. With different selections of chapters the book can serve as a text for an introductory intermediate or advanced course in quantum mechanics. Some of the sections could be used for introductions to geometrical methods in Quantum Mechanics to quantum information theory and to quantum electrodynamics and quantum field theory.