This book analyzes in considerable generality the quantization-dequantization integral transform scheme of Weyl and Wigner, and considers several phase operator theories. It features: a thorough treatment of quantization in polar coordinates, dequantization by a new method of "motes", a discussion of Moyal algebras, modifications of the transform method to accommodate operator orderings, a rigorous discussion of the Dieke laser model for one mode, fully quantum, in the thermodynamic limit, analysis of quantum phase theories based on the Toeplitz operator, the coherent state operator, the quantized phase space angle, and a sequence of finite rank operators.