We outline the quantum groups approach to the noncommutative Riemannian geometry based on quantum principal bundles and framings. Particularly the case of parallelisable quantum manifolds with nonuniversal differentials such as quantum groups and finite groups (see math.qa/0006150). Examples such as permutation group S_3 has Ricci tensor proptional the metric and (from joint work math.qa/0107216) alternating group A_4 which is Ricci-flat i.e. solves Einstein vacuum equations. If time permits, I will also mention gauge theory from math.qa/0105253 relating to flag varieties and (joint work) hep-th/0012123. Our approach also yields natural Dirac operators in the gravitational background. Among theorems, we show that all standard q-deformations of compact lie groups are Riemannian manifolds in our sense with q-Killing form metric on the underlying braided-Lie algebra. Recent results including cohomology computations will be included if time permits.