A combinatorial and field theoretic path to quantum gravity: the new challenges of group field theory

Group field theories are a new type of field theories over group manifolds and a generalization of matrix models, that have recently attracted much interest in quantum gravity research. They represent a development of and a possible link between different approaches such as loop quantum gravity and simplicial quantum gravity. After a brief introduction to the GFT formalism we put forward a long but still far from exhaustive list of open issues that this line of research faces, and that could be represent interesting challenges for mathematicians and mathematical physicists alike.