In digital control systems, the state is sampled at given sampling instants and the input is kept constant between two consecutive instants. With the optimal sampling problem, we mean the selection of sampling instants and control inputs, such that a given function of the state and input is minimized. In this paper, we formulate the optimal sampling problem and we derive a necessary condition for the optimality of a set of sampling instants in the linear quadratic regulator problem. Since the numerical solution of the optimal sampling problem is very time consuming, we also propose a new quantization-based sampling strategy that is computationally tractable and capable of achieving near-optimal cost. Finally, and probably most interesting of all, we prove that the quantization-based sampling is optimal in first-order systems for a large number of samples. Experiments demonstrate that quantization-based sampling has near-optimal performance even when the system has a higher order. However, it is still an open question whether quantization-based sampling is asymptotically optimal in any case.