For more than 70 years, quantum field theory (QFT) can be seen as a driving force in the development of theoretical physics. Equally fascinating is the fruitful impact which QFT had in rather remote areas of mathematics. The present book features some of the different approaches, different physically viewpoints and techniques used to make the notion of quantum field theory more precise. For example, the present book contains a discussion including general considerations, stochastic methods, deformation theory and the holographic AdS/CFT correspondence. It also contains a discussion of more recent developments like the use of category theory and topos theoretic methods to describe QFT. The present volume emerged from the 3rd 'Blaubeuren Workshop: Recent Developments in Quantum Field Theory', held in July 2007 at the Max Planck Institute of Mathematics in the Sciences in Leipzig/Germany. All of the contributions are committed to the idea of this workshop series: 'To bring together outstanding experts working in the field of mathematics and physics to discuss in an open atmosphere the fundamental questions at the frontier of theoretical physics'.

Constructive Use of Holographic Projections.- Topos Theory and Neo-Realist Quantum Theory.- A Survey on Mathematical Feynman Path Integrals: Construction, Asymptotics, Applications.- A Comment on the Infra-Red Problem in the AdS/CFT Correspondence.- Some Steps Towards Noncommutative Mirror Symmetry on the Torus.- Wittens Volume Formula, Cohomological Pairings of Moduli Space of Flat Connections and Applications of Multiple Zeta Functions.- Noncommutative Field Theories from a Deformation Point of View.- Renormalization of Gauge Fields using Hopf Algebras.- Not so Non-Renormalizable Gravity.- The Structure of Green Functions in Quantum Field Theory with a General State.- The Quantum Action Principle in the Framework of Causal Perturbation Theory.- Plane Wave Geometry and Quantum Physics.- Canonical Quantum Gravity and Effective Theory.- From Discrete Space-Time to Minkowski Space: Basic Mechanisms, Methods and Perspectives.- Towards a q-Deformed Quantum Field Theory.- Towards a q-Deformed Supersymmetric Field Theory.- L ?-Algebra Connections and Applications to String- and Chern-Simons n-Transport.