Most, if not all, composing is done through some form of interface, in which the composer makes modifications to, and receives feedback about, the composition in progress. Such interfaces can take a wide variety of forms, including both analog interfaces, such as pencil and paper, and digital interfaces, such as notation software and Digital Audio Workstations. This document presents a theory of these `compositional interfaces,' and of their effect on the compositional process. Starting with a review of the literature about interfaces in general, and musical interfaces more specifically, the anatomy of the compositional interface is then examined, with a detailed discussion of forms of input, representation, and feedback. Important properties of the interface as a whole---such as latency, openness, and scope---are then discussed, after which two case studies are analyzed. In the final chapter, the role of interfaces is considered within the context of the author's own work.