In this chapter we consider the averaging of periodic delay differential and delay difference equations using the method of moving averages. Specifically, we prove formal averaging theorems for both types of systems. This work is based in part on fundamental work in the averaging of delay systems performed in the 1960's by Halanay[11, 12], Hale[13], and Medvedev[23]. The analysis and theorems presented here differ from the earlier works in that our analysis gives greater importance to the delay terms which appear in the averaged system. To illustrate our results, we consider two simple examples of delay systems with periodic excitation — a cart and pendulum stabilization problem in the presence of periodic disturbances and feedback delays, and the adpative identification of chemical concentrations in a pipe mixing problem.