This monograph draws on two traditions: the algebraic formulation of quantum mechanics and quantum field theory, and the geometric theory of classical mechanics. These are combined into a unified treatment of the theory of Poisson algebras and operator algebras, based on the duality between algebras of observables and pure state spaces with a transition probability. The theory of quantization and the classical limit is discussed from this perspective. This book should be accessible to mathematicians with some prior knowledge of classical and quantum mechanics, to mathematical physicists, and to theoretical physicists who have some background in functional analysis.