Abstract: Persistence modules lie at the intersection of topological data analysis and representation theory. In applications, these structures can be used to encode the topological features of a large dataset. One of the ways to recover this topological information is to decompose the persistence module into a direct sum of indecomposable modules. In this talk, we will discuss why, even though such a decomposition is guaranteed to exist and be unique, this approach is not always feasible. We will then discuss an alternative way to study persistence modules, namely through the so-called rank invariant. Time permitting, we will conclude by discussing some recent results about both of these approaches from joint works with Benjamin Blanchette, Thomas Brüstle, and Job D. Rock.