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.