Loop Variables and Gauge Invariant Interactions in String Theory

We describe a method of writing down the exact interacting gauge invariant equations for all the modes of the bosonic open string. It is a generalization of the loop variable approach that was used earlier for the free, and lowest order interacting cases. The generalization involves, as before, the introduction of a parameter to label the different strings involved in an interaction. The interacting string has thus becomes a ``band'' of finite width. As in the free case, the fields appear to be massless in one higher dimension. Although a proof of the consistency and gauge invariance to all orders (and thus of equivalence with string theory) is not yet available, plausibility arguments are given. We also give some simple illustrations of the procedure.