<p>Recent innovations in combinatorial dynamical systems permit them to be studied with algorithmic methods. One such method from topological data analysis, called persistent homology, allows one to summarize the changing homology of a sequence of simplicial complexes. This dissertation explicates three methods for capturing and summarizing changes in combinatorial dynamical systems through the lens of persistent homology. The first places the Conley index in the persistent homology setting. This permits one to capture the persistence of salient features of a combinatorial dynamical system. The second shows how to capture changes in combinatorial dynamical systems at different resolutions by computing the persistence of the Conley-Morse graph. Finally, the third places Conley's notion of continuation in the combinatorial setting and permits the tracking of isolated invariant sets across a sequence of combinatorial dynamical systems. </p>