In this project we review the Classical and Quantum Hall Effects. We discuss theserntwo effects theoretically based on the available literature. In classical Hall Effect whenrna strong magnetic field is applied perpendicular to the electrons plane of movement,rnthe electrons execute tiny cyclotron orbits around the flux lines. If in addition, anrnelectric field applied in transverse direction to the induced electric field, the electronsrnwill tend to drift in a direction perpendicular to both fields, generating the Hall Effect.rnFor the case of Quantum Hall Effect, energy associated with the cyclotron motionrnis quantized giving rise to the Landau levels and at low temperatures all the electronsrnare in the lowest Landau level. The filling factor can be changed by varyingrnthe magnetic field B for a fixed carrier density. This leads to the sitation that thernHall conductivity takes the values equal to e2/h as discussed in Chapter three. ThernQuantum Hall Effect can be Integral or Fractional. The Integer Quantum Hall Effectrncan be understood in an independent-particle model, without taking into account thernelectron-electron interactions, in the Fractional Quantum Hall Effect, where the fillingrnfactors take fractional values (1/2,1/3,1/5,...) the electron-electron interactions playrnan essential role. The Coulomb interaction produces incompressible states of highlyrncorrelated carrier motion in high magnetic fields at specific fractional filling levels. Atrnsuch magnetic fields the electrons can be treated as quasi particles called compositernfermions