What is intrinsic and what is extrinsic in the spin Hall effect?

Two different forms of the spin Hall effect, intrinsic and extrinsic, have been recently proposed and observed in experiments. The intrinsic effect is caused by spin-orbit coupling in the band structure of the semiconductor and survives in the limit of zero disorder, whereas the extrinsic effect is caused by spin-orbit coupling between Bloch electrons and impurities. We treat both effects on equal footing within the framework of the exact Kubo linear response formalism. We show that the ``side-jump" term, which is usually considered part of the extrinsic spin Hall effect, is really intrinsic, because it is independent of disorder. Furthermore, it is the only non-zero {\it intrinsic} contribution to the spin-Hall effect for the linear Rashba (or Dresselhaus) spin-orbit coupling model. On the other hand, the skew scattering term is the only {\it extrinsic} contribution to the spin-Hall effect within this model. The proof based on gauge invariance holds at all orders in disorder and electron-electron interactions and to first order in spin-orbit coupling, but does not apply to more complex spin-orbit coupled bands (e.g the Luttinger model). We also study many-body effects and predict that the spin Coulomb drag will reduce the spin Hall conductivity.