In this paper, we discuss the existence problem of periodic solutions of the periodic difference equation x(n + 1) = f(n, x(n)),  n ∈ Z and the periodic difference equation with infinite delay x(n + 1) = f(n, xn),  n ∈ Z, where x and f are d-vectors, and Z denotes the set of integers. We show the existence of periodic solutions by using Schauder’s fixed point theorem, and illustrate an example.