Spectral Theory and Quantum Mechanics : With an Introduction to the Algebraic Formulation Introduction and mathematical backgrounds Normed and Banach spaces examples and applications Hilbert spaces and bounded operators Families of compact operators on Hilbert spaces and fundamental properties Densely defined unbounded operators on Hilbert spaces Phenomenology of quantum systems and Wave Mechanics: an overview The first 4 axioms of QM: propositions quantum states and observables Spectral Theory I: generalities abstract C algebras and operators in B(H) Spectral theory II: unbounded operators on Hilbert spaces Spectral Theory III: applications Mathematical formulation of non relativistic Quantum Mechanics Introduction to Quantum Symmetries Selected advanced topics in Quantum Mechanics Introduction to the Algebraic Formulation of Quantum Theories Order relations and groups Elements of differential geometry. This book pursues the accurate study of the mathematical foundations of Quantum Theories. It may be considered an introductory text on linear functional analysis with a focus on Hilbert spaces. Specific attention is given to spectral theory features that are relevant in physics. Having left the physical phenomenology in the background it is the formal and logical aspects of the theory that are privileged. Another not lesser purpose is to collect in one place a number of useful rigorous statements on the mathematical structure of Quantum Mechanics including some elementary yet fundamental results on the Algebraic Formulation of Quantum Theories. In the attempt to reach out to Masters or PhD students both in physics and mathematics the material is designed to be self contained: it includes a summary of point set topology and abstract measure theory together with an appendix on differential geometry. The book should benefit established researchers to organise and present the profusion of advanced material disseminated in the literature. Most chapters are accompanied by exercises many of which are solved explicitly.