  Noncommutative or `quantum' differential geometry has emerged in recent years
as a process for quantizing not only a classical space into a noncommutative
algebra (as familiar in quantum mechanics) but also differential forms, bundles
and Riemannian structures at this level. The data for the algebra quantisation
is a classical Poisson bracket, the data for the quantum differential forms is
a Poisson-compatible connection it was recently shown that after this,
classical data such as classical bundles, metrics etc. all become quantised in
a canonical `functorial' way at least to 1st order in deformation theory. There
are, however, fresh compatibility conditions between the classical Riemannian
and the Poisson structures as well as new physics such as nonassociativity at
2nd order. We give an introduction to this theory and some details for the case
of CP${}^n$ where the commutation relations have the canonical form $[w^i,\bar
w^j]=\mathrm{i}\lambda\delta_{ij}$ similar to the proposal of Penrose for
quantum twistor space. Our work provides a canonical but ultimately
nonassociative differential calculus on this algebra and quantises the metric
and Levi-Civita connection at lowest order in $\lambda$.
