### Abstract:

The study of the Hall Effect in the two-dimensional electron Systems (2DES) showed
new quantum effects. The Experimental or theoretical works show that at high magnetic
fields and low temperatures, instead of increasing linearly with increasing magnetic field
as in CHE, the Hall resistivity exhibits a series of step up increasing plateaus. In these
same intervals of magnetic fields, the longitudinal resistivity vanishes. For each plateau,
the Hall resistivity is given by the Plank’s constant h divided by the square of the
electron charge e multiplied by an integer i , which represents the number of completely
filled Landau levels. In the extreme (high magnetic fields and low temperatures) case, the
Lowest Landau Levels (LLL) is partially filled it results in FQHE where i = the fraction
number. Bits of magnetic field can get attached to each electron, creating other objects.
Such particles have properties very different from those of the electrons. They sometimes
seem to be unaware to huge magnetic fields and move in straight lines, although any bare
electron would orbit on a very tight circle. All of these strange phenomena occur in two
dimensional electron systems at low temperatures exposed to a high magnetic field only
electrons and a magnetic field. We see a brief, high light on classical Hall Effect and
overview on description of the integer and fractional quantized Hall effects, and the basic
conditions associated with their occurrence. The way in which these conditions are
satisfied, the integer quantized Hall Effect can be understood in terms of non interacting
electrons, using flux quantization. The fractional quantized Hall Effect depends
fundamentally on electron-electron interactions. Which lead to peculiar highly correlated
elementary quasi particle excitations. We shall overview and briefly discuss such
phenomena