  After review the quantum Hall effect on the fuzzy two-sphere $S^2$ and Zhang
and Hu's 4-sphere $S^4$, the incompressible quantum Hall fluid on $S^2$, $S^4$
and torus are discussed respectively. Next, the corresponding Laughlin
wavefunctions on $S^2$ are also given out. The ADHM construction on $S^4$ is
discussed. We also point out that on torus, the incompressible quantum Hall
fluid is related to the integrable Gaudin model and the solution can be given
out by the Yang Bethe ansatz.
