Quantum Independent Increment Processes II : Structure of Quantum Levy Processes Classical Probability and Physics This is the second of two volumes containing the revised and completed notes of lectures given at the school Quantum Independent Increment Processes: Structure and Applications to Physics . This school was held at the Alfried Krupp Wissenschaftskolleg in Greifswald in March 2003 and supported by the Volkswagen Foundation. The school gave an introduction to current research on quantum independent increment processes aimed at graduate students and non specialists working in classical and quantum probability operator algebras and mathematical physics. The present second volume contains the following lectures: Random Walks on Finite Quantum Groups by Uwe Franz and Rolf Gohm Quantum Markov Processes and Applications in Physics by Burkhard Kümmerer Classical and Free Infinite Divisibility and Lévy Processes by Ole E. Barndorff Nielsen Steen Thorbjornsen and Lévy Processes on Quantum Groups and Dual Groups by Uwe Franz. Random Walks on Finite Quantum Groups Quantum Markov Processes and Applications in Physics Classical and Free Infinite Divisibility and Lévy Processes Lévy Processes on Quantum Groups and Dual Groups Index.