  We establish that Hitchin's connection exist for any rigid holomorphic family
of Kahler structures on any compact pre-quantizable symplectic manifold which
satisfies certain simple topological constraints. Using Toeplitz operators we
prove that Hitchin's connection induces a unique formal connection on smooth
functions on the symplectic manifold. Parallel transport of this formal
connection produces equivalences between the corresponding Berezin-Toeplitz
deformation quantizations. - In the cases where the Hitchin connection is
projectively flat, the formal connections will be flat and we get a
symmetry-invariant formal quantization. If a certain cohomological condition is
satisfied a global trivialization of this algebra bundle is constructed. As a
corollary we get a symmetry-invariant deformation quantization.
  Finally, these results are applied to the moduli space situation in which
Hitchin's originally constructed his connection. First we get a proof that the
Hitchin connection in this case is the same as the connection constructed by
Axelrod, Della Pietra and Witten. Second we obtain in this way a mapping class
group invariant formal quantization of the smooth symplectic leaves of the
moduli space of flat SU(n)-connections on any compact surface.
