Conceptual issues for noncommutative gravity on algebras and finite sets

We discuss some of the issues to be addressed in arriving at a definitive n oncommutative Riemannian geometry that generalises conventional geometry bo th to the quantum domain and to the discrete domain. This also provides an introduction to our 1997 formulation based on quantum group frame bundles. We outline now the local formulae with general differential calculus both o n the base "quantum manifold" and on the structure group Gauge transforms w ith nonuniversal calculi, Dirac operator, Levi-Civita condition, Ricci tens or and other topics are also covered. As an application we outline an intri nsic or relative theory of quantum measurement and propose it as a possible framework to explore the link between gravity in quantum systems and entro py.