Persistent homology is a central tool in topological data analysis, used to detect geometric and topological features in data across different scales. In the classical one-parameter setting, these features are created (birth) and destroyed (death) at specific moments, closely reflecting the behavior of critical points in Morse theory. Persistence modules are obtained by applying homology functors to a filtered set of topological spaces, giving rise to representations of partially ordered sets. This allows techniques from representation theory to be applied to study questions arising in data analysis. In this minicourse, we introduce the basic theory of pointwise finite-dimensional persistence modules over a field. Topics include the classification and stability theorems for one-parameter persistence modules, as well as the associated barcode and persistence diagrams. Time permitting, we will also discuss recent developments in the theory of multiparameter persistence modules, highlighting both the new challenges that arise and the role played by representation-theoretic methods.