What is the Mathematical Structure of Quantum Spacetime?

We survey indications from different branches of Physics that the fine scale structure of spacetime is not adequately described by a manifold. Based on the hints we accumulate, we propose a new structure, which we call a quantum topos. In the process of constructing a quantum topos for quantum gravity, we propose a new, operational approach to the problem of the obervables in quantum gravity, which leads to a new mathematical point of view on the state sum models.