This century has seen the development of technologies for manipulating and controlling matter and light at the level of individual photons and atoms, a realm in which physics is fully quantum mechanical. The dominant experimental technology is the laser, and the theoretical paradigm is quantum optics. The Quantum World of Ultra-Cold Atoms and Light is a trilogy, which presents the quantum optics way of thinking and its applications to quantum devices. This book — Foundations of Quantum Optics — provides an introductory text on the theoretical techniques of quantum optics, containing the elements of what one needs to teach, learn, and “think” about quantum optics. There is a particular emphasis on the classical and quantum stochastic methods which have come to dominate the field. Book II will cover applications to quantum devices, such as quantum computers and simulators, and will include the more advanced techniques necessary to describe non-classical light fields. Book III will cover the field of ultra-cold atoms, for which the quantum-optical paradigm has proved to be highly applicable for quantitative work. Table Of Contents 。The Physical Background: >Controlling the Quantum World >Describing the Quantum World 。Classical Stochastic Methods: >Physics in a Noisy World >Stochastic Differential Equations >The Fokker–Planck Equation >Master Equations and Jump Processes >Applications of Random Processes >The Markov Limit >Adiabatic Elimination of Fast Variables 。Fields, Quanta and Atoms: >Ideal Bose and Fermi Systems >Quantum Fields >Atoms, Light and Their Interaction 。Quantum Stochastic Processes: >Quantum Markov Processes >Applications of the Master Equation 。Phase Space Methods: >Phase Space Representations for Bosons >Wigner Function Methods >P-Function Methods 。Quantum Measurement Theory: >Foundations and Formalism of Quantum Measurement >Continuous Measurements >The Quantum Zeno Effect