A simple dynamic programming argument is presented for the quadratic-cost controller synthesis problem for discrete-time linear processes with delay. Distributed delays are allowed in both state and control. The solution obtained has a discrete-time Riccati difference structure closely analogous to the Riccati differential structure associated with delay problems in continuous time. Extensions are provided for the cases of varying lag-limits, performance criterion dependent on past variables, and the time-invariant regulator problem. A feedback solution is also obtained for a continuous-time problem with distributed delays in the control, by passage to limit from the discrete results.