This work deals with the foundations of classical physics from the ''symplectic'' point of view, and of quantum mechanics from the ''metaplectic'' point of view. The Bohmian interpretation of quantum mechanics is discussed. Phase space quantization is achieved using the ''principle of the symplectic camel'', which is a recently discovered deep topological property of Hamiltonian flows. The mathematical tools developed in this book are the theory of the metaplectic group, the Maslov index in a precise form, and the Leray index of a pair of Lagrangian planes. The concept of the ''metatron'' is introduced, in connection with the Bohmian theory of motion. A precise form of Feynman's integral is introduced in connection with the extended metaplectic representation.