  In this paper we study overcomplete systems of coherent states associated to
compact integral symplectic manifolds by geometric quantization. Our main goals
are to give a systematic treatment of the construction of such systems and to
collect some recent results. We begin by recalling the basic constructions of
geometric quantization in both the Kahler and non-Kahler cases. We then study
the reproducing kernels associated to the quantum Hilbert spaces and use them
to define symplectic coherent states. The rest of the paper is dedicated to the
properties of symplectic coherent states and the corresponding Berezin-Toeplitz
quantization. Specifically, we study overcompleteness, symplectic analogues of
the basic properties of Bargmann's weighted analytic function spaces, and the
`maximally classical' behavior of symplectic coherent states. We also find
explicit formulas for symplectic coherent states on compact Riemann surfaces.
