This book provides a comprehensive overview of the probabilistic and statistical aspects of quantum theory. It is structured into six main chapters, supplemented by a supplement and references. The first chapter introduces statistical models, covering states, measurements, convex sets, and the classical and quantum statistical models. The second chapter delves into the mathematics of quantum theory, including operators, quantum states, spectral representations, and uncertainty relations. The third chapter discusses symmetry groups in quantum mechanics, focusing on Galilean relativity, kinematics, and dynamics of quantum particles. The fourth chapter addresses covariant measurements and their optimality, covering angular parameters, uncertainty relations, and estimation of pure states. The fifth chapter explores Gaussian states, including quasiclassical states, the Stone-von Neumann theorem, and characteristic functions. The sixth chapter examines unbiased measurements, including quantum communication channels, estimation of force, and bounds on measurement covariance matrices. The book also includes a supplement discussing the problem of hidden variables in quantum mechanics, and comments reflecting new results and achievements. The author emphasizes the importance of statistical methods in quantum theory, particularly in quantum estimation theory, and highlights the role of mathematical statistics in understanding quantum measurements. The book is intended for a broad audience of mathematicians and physicists, providing a mathematical foundation for quantum theory while emphasizing its statistical aspects. It is not a standard textbook on quantum mechanics but rather a specialized work on the probabilistic and statistical foundations of quantum theory. The author acknowledges the contributions of various scholars and emphasizes the importance of quantum information theory and quantum computation in the development of the field. The book is written in a mathematical style but is accessible to a wide audience, with a focus on clarity and practical applications.This book provides a comprehensive overview of the probabilistic and statistical aspects of quantum theory. It is structured into six main chapters, supplemented by a supplement and references. The first chapter introduces statistical models, covering states, measurements, convex sets, and the classical and quantum statistical models. The second chapter delves into the mathematics of quantum theory, including operators, quantum states, spectral representations, and uncertainty relations. The third chapter discusses symmetry groups in quantum mechanics, focusing on Galilean relativity, kinematics, and dynamics of quantum particles. The fourth chapter addresses covariant measurements and their optimality, covering angular parameters, uncertainty relations, and estimation of pure states. The fifth chapter explores Gaussian states, including quasiclassical states, the Stone-von Neumann theorem, and characteristic functions. The sixth chapter examines unbiased measurements, including quantum communication channels, estimation of force, and bounds on measurement covariance matrices. The book also includes a supplement discussing the problem of hidden variables in quantum mechanics, and comments reflecting new results and achievements. The author emphasizes the importance of statistical methods in quantum theory, particularly in quantum estimation theory, and highlights the role of mathematical statistics in understanding quantum measurements. The book is intended for a broad audience of mathematicians and physicists, providing a mathematical foundation for quantum theory while emphasizing its statistical aspects. It is not a standard textbook on quantum mechanics but rather a specialized work on the probabilistic and statistical foundations of quantum theory. The author acknowledges the contributions of various scholars and emphasizes the importance of quantum information theory and quantum computation in the development of the field. The book is written in a mathematical style but is accessible to a wide audience, with a focus on clarity and practical applications.