  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.
