We study discrete almost automorphic functions (sequences) defined on the set of integers with values in a Banach space X . Given a bounded linear operator T defined on X and a discrete almost automorphic function f ( n ) , we give criteria for the existence of discrete almost automorphic solutions of the linear difference equation Δ u ( n ) = T u ( n ) + f ( n ) . We also prove the existence of a discrete almost automorphic solution of the nonlinear difference equation Δ u ( n ) = T u ( n ) + g ( n , u ( n ) ) assuming that g ( n , x ) is discrete almost automorphic in n for each x ∈ X , satisfies a global Lipschitz type condition, and takes values on X .