In trying to get my head round GR and quantum gravity, I'm puzzled about the following questions: Is the guage group for gravity defined as the set of all possible Weyl tensors on a general 4D Riemann manifold? Which abstract group maps onto this set? Is it GL(4) or a subgroup of GL(4)? How do you derive the number of gravitational force bosons from the guage group structure? Which abstract groups represent all possible Riemann curvature tensors, and all possible metric tensors? What is the equivalent of the Lorentz group for GR? I.e. the group of transformations between all possible reference frames. How is all of this connected with the conformal group? What is the purpose of conformal invariance?