The relation of quantum and classical phenomena has been a subject of continuing interest. Most studies approximate the quantum description of a phenomenon to obtain a classical or semi-classical approximation. This book develops a new formalism that contains both fully quantum and classical sectors, and a continuous transformation between them that provides an intermediate partly quantum - partly classical sector. This intermediate sector can play the role of a bridge between the quantum and classical descriptions of a process. Using this new formalism we consider the case of the harmonic oscillator in detail relating the quantum oscillator through the bridge to the classical oscillator. We then develop a generalization of the Feynman path integral formalism that has both a normal quantum sector and also a 'new' classical path integral sector - again with a partly quantum-partly classical intermediate sector. Our path integral generalization yields a generalization of the Schroedinger equation with both quantum and classical 'wave function' solutions. We also apply this formalism to the Fokker-Planck equation, for which it is naturally adapted. Next we apply the new formalism to quantum field theories that are known to be chaotic, and then generate a classical sector - also with chaotic behavior. We also take the standard approach to quantum entanglement and show how to extract a semi-classical entanglement as well as a classical limit without entanglement. The Boltzmann equation is easily placed within the framework of our new formalism. We solve a special case of the Vlasov equation as an example. A special relativistic Boltzmann equation is also developed within the framework of our formalism. Lastly, we develop our formalism for boson and fermion quantum field theory. We give a sensible reason why Nature must be quantum.