This is a monograph in mathematics, volume 85, edited by H. Amann, K. Grove, H. Kraft, and P.-L. Lions. It includes associate editors from various universities and institutions. The book is titled "An Introduction to Quantum Stochastic Calculus" by K.R. Parthasarathy, published by Springer Basel AG. It includes bibliographical references and is cataloged in both the Library of Congress and the Deutsche Bibliothek. The book is a revised version of mimeographed notes and is dedicated to Brahma, Vishnu, and Shiva, representing creation, protection, and destruction. The book explores quantum probability within the framework of operators and group representations in Hilbert space. It begins with an introduction to quantum probability, covering topics such as events, observables, states, and dynamics in finite-dimensional quantum probability spaces. It then delves into tensor products of Hilbert spaces, symmetric and antisymmetric tensor products, and the Weyl representation, which is a summary of the Weyl commutation relations and the second quantization homomorphism. The book also discusses creation, conservation, and annihilation operators, which are fundamental in quantum mechanics. The second chapter explores the relationship between infinitely divisible probability distributions and the projective unitary Weyl representation of the Euclidean group of a Hilbert space. The third chapter presents a quantum stochastic calculus based on the notions of creation, conservation, and annihilation operators, leading to quantum Ito's formula. It highlights the connection between classical stochastic processes and quantum stochastic processes, and discusses quantum dynamical semigroups, quantum stochastic flows, and classical Markov chains in the quantum framework. The book also addresses the challenges and limitations of the subject, including the rapid growth of literature and the need for more physical and mathematical examples. It acknowledges the contributions of various scholars and the importance of quantum stochastic calculus in understanding dynamical phenomena. The monograph is a comprehensive resource for those interested in quantum probability and stochastic calculus.This is a monograph in mathematics, volume 85, edited by H. Amann, K. Grove, H. Kraft, and P.-L. Lions. It includes associate editors from various universities and institutions. The book is titled "An Introduction to Quantum Stochastic Calculus" by K.R. Parthasarathy, published by Springer Basel AG. It includes bibliographical references and is cataloged in both the Library of Congress and the Deutsche Bibliothek. The book is a revised version of mimeographed notes and is dedicated to Brahma, Vishnu, and Shiva, representing creation, protection, and destruction. The book explores quantum probability within the framework of operators and group representations in Hilbert space. It begins with an introduction to quantum probability, covering topics such as events, observables, states, and dynamics in finite-dimensional quantum probability spaces. It then delves into tensor products of Hilbert spaces, symmetric and antisymmetric tensor products, and the Weyl representation, which is a summary of the Weyl commutation relations and the second quantization homomorphism. The book also discusses creation, conservation, and annihilation operators, which are fundamental in quantum mechanics. The second chapter explores the relationship between infinitely divisible probability distributions and the projective unitary Weyl representation of the Euclidean group of a Hilbert space. The third chapter presents a quantum stochastic calculus based on the notions of creation, conservation, and annihilation operators, leading to quantum Ito's formula. It highlights the connection between classical stochastic processes and quantum stochastic processes, and discusses quantum dynamical semigroups, quantum stochastic flows, and classical Markov chains in the quantum framework. The book also addresses the challenges and limitations of the subject, including the rapid growth of literature and the need for more physical and mathematical examples. It acknowledges the contributions of various scholars and the importance of quantum stochastic calculus in understanding dynamical phenomena. The monograph is a comprehensive resource for those interested in quantum probability and stochastic calculus.