For controlled Rⁿ-valued linear systems driven by Gaussian noise under quadratic cost criteria, we investigate the existence and the structure of optimal quantization and control policies. For fully observed and partially observed systems, we establish the global optimality of a class of predictive encoders and show that an optimal quantization policy exists, provided that the quantizers allowed are ones which have convex codecells. Furthermore, optimal control policies are linear in the conditional estimate of the state, and a form of separation of estimation and control holds.