The Structure and Stability of Persistence Modules This book is a comprehensive treatment of the theory of persistence modules over the real line. It presents a set of mathematical tools to analyse the structure and to establish the stability of such modules providing a sound mathematical framework for the study of persistence diagrams. Completely self contained this brief introduces the notion of persistence measure and makes extensive use of a new calculus of quiver representations to facilitate explicit computations. Appealing to both beginners and experts in the subject The Structure and Stability of Persistence Modules provides a purely algebraic presentation of persistence and thus complements the existing literature which focuses mainly on topological and algorithmic aspects Introduction Persistence Modules Rectangle Measures Interleaving The Isometry Theorem Variations References Index