In this book we analyze the modern state of the problem connected with quantum Hall effect (integral and fractional). The quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two dimensional electron systems subjected to low temperatures (> 1K) and strong magnetic fields (10T), in which the Hall conductance takes on the quantized values (e2/h) with an integer (integer quantized Hall effect) or a rational fraction (fractional quantized Hall effect), independent of the detail of the sample geometry. The fractional quantum Hall effect is not complectly understood at the time being. Recently the idea of chiral heat transport in quantum Hall regime was pushed forward , we also analyze thermal transport in the fractional quantum Hall effect (FQHE). And we consider the problem of two electrons in a uniform magnetic field with the account of the center mass: (1) Quantum Mechanical Description and (2) Classical Description.