In the first part of this dissertation, we study mainly novel boron structures and their electronic and mechanical properties, using ab initio calculations. The electronic structure and construction of the boron buckyball B80, and boron nanotubes as the α-sheet wrapped around a cylinder are studied. The α-sheet is considered so far to be the most stable structure energetically out of the two dimensional boron assemblies. We will argue however that there are other sheets close in energy, using cluster expansion method. The boron buckyball is shown to have different possible isomers. Characterization of these isomers according to their geometry and electronic structure is studied in detail. Since the B80 structure is made of interwoven double-ring clusters, we also investigate double-rings with various diameters. We investigate the properties of nanotubes obtained from α-sheet. Computations confirm their high stability and identify mechanical stiffness parameters. Careful relaxation reveals the curvature-induced buckling of certain atoms off the original plane. This distortion opens up the gap in narrow tubes, rendering them semi-conducting. Wider tubes with the diameter d  1.7 nm retain original metallic character of the α-sheet. We conclude this part by investigation into hydrogen storage capacity of boron-rich compounds, namely the metallacarboranes. In the second part of dissertation, we switch our focus to electronic and transport properties of carbon nano-structures. We study the application of carbon nanotubes as electro-chemical gas sensors. The effect of physisorption of NO2 gas molecules on electron transport properties of semi-conducting carbon nanotubes is studied using ab initio calculations and Green’s function formalism. It is shown that upon exposure of nanotube to different concentrations of gas, the common feature is the shift in conductance towards lower energies. This suggests that physisorption of NO2 will result in a decrease (increase) in conductance of p-type (n-type) nanotubes with Fermi energies close to the edge of valence and conduction band. Finally we study the effect of torsion on electronic properties of carbon nano-ribbons, using helical symmetry of the structures.