Modern electronics technology has reached a very sophisticated stage. Requirements for smaller and reliable devices are becoming more and more demanding. It is predicted that Moore's law in making a faster CPU is going to break down in the foreseeable future. People start to look into new materials in the hope that soon in the future, one can replace silicon in fabricating computer chips. Many novel properties arise in meso- and nanoscale systems, among which electron transport properties of such systems are of particular interest to us. Due to the small sizes of the systems, a full quantum mechanical description of the electrons is required. In general, materials used in electron transport studies can be categorized into three kinds: (i) single molecular systems, which have well-defined finite molecular structures, and well quantized energy levels, including pi conjugated molecules, such as benzene-dithiol, beta-carotene and metal-ligand clusters, such as cobalt bis(terpyridyl) complex and a lot more; (ii) extended "molecules" with repeated structures, such as carbon nanotubes and graphene sheets. These systems typically show continuous energy dispersions in one or more dimensions and quantized subbands in the other dimensions; (iii) quantum dots, which are small fragments of crystalline semiconductors, such as CdSe or Si, with many thousands of atoms. The quantum effects in such systems are usually tuned by adjusting the sizes of these dots. In this thesis, we used theoretical tools such as the equation of motion Green's function technique, non-equilibrium Green's functions and linear response theory to try to understand the physics underlying the electron transport processes in some of the systems mentioned above. In particular, we used an equation of motion Green's function technique combined with the non-equilibrium Green's function formalism to study the effects arising from the electron-electron interaction in the electron transport process. In model calculations we found extra structure for the current-voltage relation at higher biases as compared to the one calculated at a mean-field level. We also studied the Coulomb blockade problem that has been experimentally realized in molecular systems such as cobalt bis(terpyridyl) complex where the effects of gating and bias voltage play an important role. The correct description of electron self-interaction effects were found to be crucial to reproducing the correct Coulomb blockade behaviour. For materials with extended structure, we studied the effects of various static scattering sources on the transmission of electrons through graphene nanoribbons, and further the effects of gating, in particular, in relation to the so-called Klein paradox in graphene. Here we found an interesting subband dependent scattering process that led to significant changes in the transmission as a function of the angle of the applied gate. We also studied the effect of dynamical scattering due to electron-electron interactions in graphene nanoribbons, which were found to lead to large changes in the quantitative transmission coefficients.