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.

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).