Quantum mechanics - and its relativistic version: quantum field theory (QFT) - is immensely successful and unrivaled since 80 years. Yet its principles and its seemingly paradoxical aspects are further tested, and actively discussed and commented. The goal of these lectures is to present an introductory but hopefully coherent view of the main formalizations of quantum mechanics, of their interrelations, and of their theoretical foundation. The course is aimed at a non-specialized audience: graduate students and more advanced researchers, not necessarily theorists. The mathematical formalism will be presented and discussed at a simple and not-too-abstract level. This course will not attempt to cover the historical and philosophical aspects of quantum physics. The tentative plan of the lectures follows. \ 1 - Reminders: \ hspace*{0.2cm} textdegree Classical mechanics: states, observables & probabilities \ hspace*{0.2cm} textdegree The canonical and path integral formulations of quantum mechanics & QFT \ hspace*{0.2cm} textdegree Causality, reversibility & locality \ 2 - Algebraic quantum theory: \ hspace*{0.2cm} textdegree The algebra of observables \ hspace*{0.2cm} textdegree States & real C*-algebras \ hspace*{0.2cm} textdegree The GNS constructions, complex Hilbert spaces \ hspace*{0.2cm} textdegree Algebraic QFT & von Neumann algebras \ 3 - ``Quantum logic'' formulations: \ hspace*{0.2cm} textdegree The lattice of propositions & orthomodular geometry \ hspace*{0.2cm} textdegree Soler's & Gleason's theorems, Hilbert spaces (again) \ 4 - Quantum information (operational) formulations \ 5 - A few words (depending on time) on: \ hspace*{0.2cm} textdegree Quantum correlations: causality, non-localities and contextuality \ hspace*{0.2cm} textdegree Quantum measurements & the quantum-to-classical transition \ hspace*{0.2cm} textdegree Interpretations of quantum mechanics versus alternative quantum theories \ ... to say nothing of quantum gravity ... \ \ \ (Cours organisés en collaboration avec l'Ecole Doctorale de Physique de la Région Parisienne - ED 107)