Functional composition, the application of one function to the results of another function, has a long history in the mathematics community, particularly in the context of polynomials and rational functions. This thesis articulates and explores a general framework for the use of functional composition in the context of signal processing. Its many potential applications to signal processing include utilization of the composition of simpler or lower order subfunctions to exactly or approximately represent a given function or data sequence. Although functional composition currently appears implicitly in a number of established signal processing algorithms, it is shown how the more general context developed and exploited in this thesis leads to significantly improved results for several important classes of functions that are ubiquitous in signal processing such as polynomials, frequency responses and discrete multivariate functions. Specifically, the functional composition framework is exploited in analyzing, designing and extending modular filters, separating marginalization computations into more manageable subcomputations and representing discrete sequences with fewer degrees of freedom than their length and region of support with implications for sparsity and efficiency.