Quantum Field Theory and Gravity : Conceptual and Mathematical Advances in the Search for a Unified Framework Preface Quantum Gravity: Whence Whithter (Claus Kiefer) Local Covariance and Background Independence (Klaus Fredenhagen Katarzyna Rejzner) The Big Wave Theory for Dark Energy (Blake Temple). Discrete and Continuum Third Quantization of Gravity (Steffen Gielen Daniele Oriti) Unsharp Values Domains and Topoi (Andreas Döring Rui Soares Barbosa) Causal Boundary of Spacetimes: Revision and Applications to AdS CFT Correspondence (José Luis Flores Jónatan Herrera Miguel Sánchez) Some Mathematical Aspects of the Hawking Effect for Rotating Black Holes (Dietrich Häfner) Observables in the General Boundary Formulation (Robert Oeckl) Causal Fermion Systems: A Quantum Space Time Emerging From an Action Principle (Felix Finster Andreas Grotz Daniela Schiefeneder) CCR Versus CAR Quantization on Curved Spacetimes (Christian Bär Nicolas Ginoux) On the Notion of the Same Physics in All Spacetimes (Christopher J. Fewster) Local Covariance Renormalization Ambiguity and Local Thermal Equilibrium in Cosmology (Rainer Verch) Shape Dynamics. An Introduction (Julian Barbour) On the Motion of Point Defects in Relativistic Fields (Michael K. H. Kiessling) How Unique are Higher dimensional Black Holes (Stefan Hollands) Equivalence Principle Quantum Mechanics and Atom interferometric Tests (Domenico Giulini). One of the most challenging problems of contemporary theoretical physics is the mathematically rigorous construction of a theory which describes gravitation and the other fundamental physical interactions within a common framework. The physical ideas which grew from attempts to develop such a theory require highly advanced mathematical methods and radically new physical concepts. This book presents different approaches to a rigorous unified description of quantum fields and gravity. It contains a carefully selected cross section of lively discussions which took place in autumn 2010 at the fifth conference Quantum field theory and gravity Conceptual and mathematical advances in the search for a unified framework in Regensburg Germany. In the tradition of the other proceedings covering this series of conferences a special feature of this book is the exposition of a wide variety of approaches with the intention to facilitate a comparison. The book is mainly addressed to mathematicians and physicists who are interested in fundamental questions of mathematical physics. It allows the reader to obtain a broad and up to date overview of a fascinating active research area.