During the past decade two-dimensional electron systems at low temperatuer and highrnmagnetic eld have repeatedly surprised with exotic electron correlation phenomena. Thernquantum liquids of the fractional quantum Hall e ect (FQHE), the still enigmatic electronrncrystal at very low lling factors, and vanishing and reappearance of certain quantumrnHall states in double layer electron systems all rernect the dominance of electron-electronrninteraction at very high magnetic elds. Recently the nature of the electronic interactionrnat the half- lled Landau levels has received much attention. The magnetization of twodimensionalrnelectron gases at low temperatures and in a strong magnetic eld B orientedrnnormal to the plane of the electron motion was calculated along the same lines as in thernwork of Landau. With in two-dimensional electron system new surprises arise year afterrnyear, exotic states of electronic matter, new materials with fascinating quantum Hallrnproperties keep triggering an intense activity in the eld. Two-dimensional electron gasesrnwith crossing energy levels can be realized in a variety of systems with di erent relativernsizes of orbital and spin e ects, Coulomb energy, and di erent coupling between the components.rnTheir study has lead to the discovery of many correlated Quantum Hall statesrnand much e ort is put into unravelling the energy-level structure of these systems. In thisrnwork we have four parts. The rst part contens Two-Dimensional Electron gas, QuantumrnCon nement and Density of States, Potential wells in Semiconductors, Con nement ofrnElectrons, E ect of Quantum Con nement on Electrons, Semiconductors Nanostructuresrnwith Con ned Electrons and Properties and Applications of Con ned Electron Gases.rnThe second part includes Landau Levels and Landau Level degeneracy. The third partrnis about Classical and Quantum Hall e ect. Finally the forth part presents two chargedrnparticles in a magnetic eld and Parabolic con nement in strong magnetic eld. Here wernsee charged particle in weak magnetic eld and how the particle is confi ned