Given a complex summable sequence a(n) and a discrete almost automorphic function f (n) with values in a complex Banach space X, we give criteria for the existence of discrete almost automorphic solutions of the linear Volterra difference equation u(n + 1) = lambda Sigma(n)(j=-infinity) a(n - j)u(j) + f(n), n is an element of Z, for lambda in a distinguished subset of the complex plane, determined by a(n). We also prove the existence of a discrete almost automorphic solution of the nonlinear Volterra difference equation u(n + 1) = lambda Sigma(n)(j=infinity) a(n - j)u(j) + f (n, u(n), n is an element of Z, where f (n, x) is a discrete almost automorphic function in n for each x [ X that satisfies a global Lipschitz condition and takes values in X. Finally, we treat the same type of problemfor Volterra functional difference equations.