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We present in this paper some of the relationships and applications of the operator model theory of Sz-Nagy and Foias to system and network theory. The three basic objects of the Nagy-Foias theory are: a special Hilbert space called the Nagy-Foias space, an operator on this space called the compressed shift operator, and the characteristic operator function of a contraction operator. It will be shown that these objects fit naturally in the state space realization of linear systems, and in the scattering synthesis of linear passive networks. We conclude the paper with a brief discussion of the Jordan model for a class of operators, and some of its potential applications to systems and networks.