  An elementary introduction is provided to the phase space quantization method
of Moyal and Wigner. We generalize the method so that it applies to
2-dimensional surfaces, where it has an interesting connection with quantum
holography. In the case of Riemann surfaces the connection between Moyal
quantization and holography provides new insights into the Torelli theorem and
the quantization of non-linear integrable models. Quantum holography may also
serve as a model for a quantum theory of membranes.
