A general theory of "interface responses" in discrete composite d-dimensional systems for operators with two-body interactions is presented. It is shown that the "interface responses" of all the internal and external interfaces of any composite system are the linear superposition of the responses to a coupling operator of all individual interfaces and of the responses to a cleavage operator of the corresponding ideal free surfaces of the same but non-interacting subsystems. The response function and its elements between two space points of the system are given by a new simple general equation as a function of these "interface responses" and of the bulk response functions of each subsystem contained in the complete real system. The present paper establishes this new general two-body theory of interface responses for surfaces, interfaces, adsorbates, membranes, superlattices, defects of any kind and dimension, ... and for the first time, to the knowledge of the author, for any d-dimensional composite system. The presentation of the theory is followed here by a few general applications.