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Together with collaborators, we introduced a noncommutative Riemanniangeometry over Moyal algebras and systematically developed it for noncommutativespaces embedded in higher dimensions in the last few years. The theory wasapplied to construct a noncommutative version of general relativity, which isexpected to capture some essential structural features of spacetime at thePlanck scale. Examples of noncommutative spacetimes were investigated indetail. These include quantisations of plane-fronted gravitational waves,quantum Schwarzschild spacetime and Schwarzschild-de Sitter spacetime, and aquantun Tolman spacetime which is relevant to gravitational collapse. Here webriefly review the theory and its application in the study of quantum structureof spacetime.

A noncommutative geometric approach to the quantum structure of spacetime