  This is a self-contained introduction to quantum Riemannian geometry based on
quantum groups as frame groups, and its proposed role in quantum gravity. Much
of the article is about the generalisation of classical Riemannian geometry
that arises naturally as the classical limit; a theory with nonsymmetric metric
and a skew version of metric compatibilty. Meanwhile, in quantum gravity a key
ingredient of our approach is the proposal that the differential structure of
spacetime is something that itself must be summed over or `quantised' as a
physical degree of freedom. We illustrate such a scheme for quantum gravity on
small finite sets.
