All classical (abelian) theta functions, first studied by Jacobi and Riemann, satisfy the heat equation on a torus. When this torus is the Jacobian of a Riemann surface X, these theta functions have special properties reflecting the geometry of the surface. Since the non-abelian analogue of the Jacobian is the moduli space of vector bundles over X, non-abelian theta functions are defined as holomorphic sections on this moduli space. However, the analytic theory for these functions is still under investigation. We will show that, in some cases, all abelian and non-abelian theta functions can be obtained by a certain extension of the coherent state transform on a Lie group, which naturally induces the inner product on the space of theta functions.