Structure at infinity for systems with delays is considered. A generalization of the Smith-McMillan form at infinity is given with an application to the Model Matching Problem. With the help of some finite dimensional descriptions, a formulation of a Partial Disturbance Decoupling Problem is given for systems with delay and necessary and sufficient structural conditions are provided for the existence of a non anticipative stale feedback solution to this problem. These conditions arc expressed in terms of structures at infinity and directly extend the corresponding result for Exact Disturbance Decoupling for classical systems without delay.