A new model of abstract automata is presented employing the concept of finite automata on a network. Each normal network n provided with a one-way input tape determines a family of languages nl. A representation theorem, analogous to the Chomsky-Schutzenberger representation theorem for context free languages1, is proved for the class nl. One consequence is that nl is a principal full AFL generated by a closed set (one that contains all its prefixes). The converse is also proved, thereby establishing an equivalence between families of languages defined by normal networks and principal full AFLs generated by closed sets. The representation theorem is applied to the push-down store and Turing machine networks to obtain a stronger version of the Ginsburg, Greibach, and Harrison representation theorem for recursively enumerable sets6.