In recent years semiconductor nanostructures have become the model systems of choice for investigations of electrical conduction on short length scales. Quantum traort is conveniently studied in a two dimensional electron gas because of the combination of a large Fermi wavelength and large mean free path. The phase coherence of charge carriers gives rise to the unique traort roperties of mesoscopic systems. This makes them interesting to study from a fundamental point of view, but also gives these small systems a possible future in nanoelectronics applications. In the present work, a numerical method is implemented in order to contribute to the understanding of quantum traort in narrow channels that electrons have confined by an electrostatic potential and therefore form a quasi one-dimensional system. we introduce an approach that has proved to be very useful in describing mesoscopic traort. In this approach, the current through a conductor is expressed in terms of the probability that an electron can transmit through it. Then we describe Büttiker extension on this approach to describe multi-terminal measurement in magnetic fields (generally referred to as the Landauer- Büttiker formalism). To simplify the discussion, we assume zero temperature and phase coherent traort. We introduce the concept of Green's functions and then show that a conductor connected to infinite leads can be replaced by a finite conductor with the effect of the leads incorporated through a 'self-energy' function. This provides a convenient method for evaluating the Green's function (and hence the transmission function) numerically. A numerical technique, for calculating the transmission coefficients through a coherent mesoscopic conductor using it's green's function is introduced. The conductance of phase coherent quasi one dimensional systems is studied in ballistic, diffusive and high field regimes . Our ;calculations was done in a model system with Hard-wall boundary conditions in the transverse direction and the Anderson model of disorder was used in disordered samples. Spin splitting was ignored and spin degeneracy was assumed. We have presented the results of quantum traort for different strengths of disorder and introduced magnetic fields. Our results confirmed the Landauer formalism for calculation of electronic traort. We observed conductance fluctuations due to quantum interference in our numerical results and showed that weak localization effect can be remove by application of a weak perpendicular magnetic field. Magnetic depopulation of subbands in ballistic samples was studied numerically. Finally we showed numerically the transition to the integral quantum Hall effect regime through the uppression of ackscattering on a disordered model system y calculating the two terminal conductance of a quasi-one-dimensional quantum conductor as a strong magnetic field is applied. Our results showed that this regime is entered when there is negligible overlap between electron edge states localized at opposite sides of the ample. Key words : quantum traort , Green’s function, tight-binding model, disorder, magnetotraort .