Appears in collection : Mathematics on the Crossroad of Centuries - A Conference in Honor of Maxim Kontsevich's 60th Birthday

We will consider Geometric Quantization on general Kahler phase spaces and propose a program for compatible constructions of the quantization of functions, the Hilbert space structure, and the dependence on the choice of the Kahler structure (generalized Hitchin connection), fixing only the underlying symplectic manifold and a prequantum line bundle. We will in particular see explicitly how the curvature of the phase space modifies the quantization.