I shall look at two factors that dominate the quantum Hall system. One is the many-body, fractional quantum Hall system. Depending on the range of the electron-electron interaction projected onto the Landau level, various quantum phases are proposed, including non-Fermi liquids and paired states of composite fermions. The other factor is the spatial dimensionality. While usually the quantum Hall effect is specific to 2D, we propose an integer QHE should be possible in 3D if the shape of the Fermi surface fulfills certain conditions. One such case is the field-induced SDW in incompletely-nested (i.e., multiply-connected) Fermi surfaces, so the interplay of many-body physics and Landau’s quantization on various Fermi surfaces may provide an interesting future avenue.

University of Tokyo, Hongo