This dissertation proposes cellular sheaf theory as a method for decomposing data analysis problems. We present novel approaches to problems in pursuit and evasion games and topological data analysis, where cellular sheaves and cosheaves are used to extract global information from data distributed with respect to time, boolean constraints, spatial location, and density. The main contribution of this dissertation lies in the enrichment of a fundamental tool in topological data analysis, called persistent homology, through cellular sheaf theory. We present a distributed computation mechanism of persistent homology using cellular cosheaves. Our construction is an extension of the generalized Mayer-Vietoris principle to filtered spaces obtained via a sequence of spectral sequences. We discuss a general framework in which the distribution scheme can be adapted according to a user-specific property of interest. The resulting persistent homology reflects properties of the topological features, allowing the user to perform refined data analysis. Finally, we apply our construction to perform a multi-scale analysis to detect features of varying sizes that are overlooked by standard persistent homology.