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 group of all possible Weyl tensors on a general 4D Riemann manifold? How is this group defined in matrix algebra? Is it a subgroup of GL(4). How do you derive the number of gravitational force bosons from the group structure? What 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?