The Hall effect was discovered by Edwin Hall in 1879 when he was a graduate student in the Johns Hopkins University under the advisory of Professor Henry A. Rowland, after whose name this department is named now. But at that time, even the electron was not experimentally discovered. Clear understanding had to wait until quantum mechanics came into apperance.
In 1930, Landau showed that for quantum electrons, unlike classical electrons, the electron's orbital motion gave a contribution to the magnetic susceptibility. He also remarked that the kinetic energy quantization gave rise to a contribution to the magnetic susceptibility which was periodic in inverse magnetic field. We can see later that Landau levels along with localization can explain the integer quantum Hall effect satisfactorily.
The first inversion layer Hall conductivity measurements in strong magnetic fields were done by S.Kawaji and his colleagues in 1975. Using a somewhat different experimental arrangement which measured the Hall voltage rather than the Hall current, Klaus von Klitzing and Th. Englert had found flat Hall plateaus in 1978. However, the precise quantization of the Hall conductance in units of 
was not recognized until February of 1980. Five years later, in 1985, Klaus von Klitzing was awarded Nobel Prize in Physics for the discovery of quantum Hall effect.
This was not the end of the story. In 1982 D.C.Tsui, H.L.Störmer, and A.C.Gossard discovered the existance of Hall steps with rational fractional quantum numbers, which is called fractional quantum Hall effect. R.B.Laughlin's wave functions established a very good, though not yet perfect understanding of this phenomenon. Today, the study of quasiparticles of fractional charge and fractional statistics are still active areas of research.