Carbon nanotubes (CNTs), quasi-one-dimensional structures rolled from the twodimensional hexagonal lattices of graphene sheets with the local bond arrangement conserved, provide opportunities for engineering novel physical properties suitable for ultimate device miniaturization. Recent interest in CNTs has focused on the various physical properties of nanotubes functionalized in different ways. The presence of weak interactions, e.g. van der Waals interaction and π-π stacking, is known to allow CNTs to aggregate into bundles and to make CNTs soluble in organic solvents. Modification of CNT surfaces with functional molecules is expected to give rise to novel materials with interesting properties and prospects for future nanodevice applications. In the thesis, a conventional density-functional theoretical method and a selfconsistent- charge density-functional tight-binding (SCC-DFTB) approach augmented by an empirical London dispersion correction (DFTB-D), were used to study the carbon based nanotubes modified with various functional molecules. Previous works have demonstrated that the DFTB-D method is an efficient way to deal with systems involving weak interactions. In Chapter 3, noncovalent exo-modifications of CNTs with conjugated polymer were explored. Noncovalent surface modification is expected to open up the possibility of organizing CNTs into ordered networks without disrupting their electron conjugation characteristics. Our calculated results showed that the binding energies between poly phenylenethynylene (PPE) and (n, n) CNTs increase when the diameters increase for n≤9 and then decrease due to steric hindrance effect. It is evident that the combination is made by π - π and van der Waals interactions with very small charge transfers involved. Significant blue shifts in the absorption and emission spectra of PPE upon interacting with the CNTs were revealed in the calculation, which are responsible to the optical quenching observed in experiment. The mechanism of water transport through membrane channels has attracted great attention but still remains poorly understood because of the difficulty in separating generic causative features of transport from specific molecular and chemical details. Due to the similarities in the structures of the hydrophobic channels of transmembrane protein with CNTs, the simulated process of water transporting in CNTs is an excellent model for understanding water transport through membrane channels. In Chapter 4, the water chains encapsulated within CNTs were studied. The most interesting and important feature we observed is the diameter shrinking of CNTs when water chains are confined inside CNTs. The diameter shrinking of CNTs can be attributed to the van der Waals and H-π interaction between water chains and CNTs. Our calculated Raman spectra showed that the interactions between CNTs and water chains probably give rise to a kind of “mode hardening effect,” consistent with the diameter shrinkage of CNTs when water chains are confined in CNTs. The calculated hydrogen bonding energy is 2.64 kcal/mol, suggesting the presence of weak hydrogen bonding when water is confined in the narrow tube. Due to the high surface area to volume ratio, CNTs have been proposed to be a kind of possible hydrogen storage materials. In Chapter 5, we performed first-principles calculations based on DFT to study the electronic and atomic structures of fully hydrogenated CNTs which are relevant to hydrogen storage and device application. Our calculations showed that the exohydrogenated C atoms strongly stick out and the endhydrogenated C atoms moved strongly inwards, resulting in an inner endhydrogenated C cylinder and an outer exodhydrogenated C cylinder. The hydrogenation energy varies linearly with the inverse of the square of the tube diameter. The armchair structure has a larger binding energy than the zigzag structure, for a given diameter of fully hydrogenated CNTs (CnHn), suggesting that hydrogenation is more exothermic for the armchair nanotubes for a given tube radius. The band gap of CNTs is greatly increased by the hydrogenation of all carbon atoms of CNTs. The observed band gap opening via hydrogenation of nanotubes can be used in band gap engineering for device applications. There exist all kinds of different length of the ultrasmall radius single-walled nanotubes grown from zeolite templates (e.g., AFI) by pyrolysis of tripropylamine (TPA) moleculs. In Chapter 6, we employed the density-functional tight-binding method to study the tube length effect on the structural, electronic, vibrational, and optical propterties of ultrasmall radius single-walled nanotubes. The successful synthesis of BN nanotube has open up the possibility of the existence of tubular structures made of either noncarbon or partially carbon materials. For example, silicon carbide nanotubes (SiCNTs) have been produced via the reaction of Si with carbon nanotubes. SiCNTs are expected to have the advantages over CNTs because they may possess high reactivity of exterior surface facilitating sidewall decoration and stability at high-temperature. There are two possible structures for SiCNTs: one consists of alternating Si and C atoms and another one consists of pairs of C=C and Si=Si. In the thesis (Chapters 7 and 8), the two distinct types of SiCNT have been studied. Our results showed that the two distinct categories are close in energies, but show significant difference in electronic and transport properties. For the first type of structure (alternating Si and C atoms), both the cohesive and the strain energies vary inversely proportional to the square of the nanotube diameter, irrespective of the type (zigzag, armchair or chiral) of the SiCNTs. In contrast, the energy gap is very sensitive to both the diameter and the type of the nanotube. All three types of SiCNTs (zigzag, armchair and chiral) were found to be semiconductors with small band gaps (~1eV). However, the zigzag and chiral SiCNTs considered here show n = 3k anomaly, with the band gaps of zigzag (n, 0) SiNTs being somewhat larger and approaching those of the armchair (n, n) SiCNT when n = 3k. For the second type of structure (pairs of C=C and Si=Si), the SiCNTs structures exhibit considerable distortions with one of the silicon atoms of the Si=Si bond being sp3-like while the remaining silicon atom and the C=C bond being sp2- like. As a result, the tube cross sections adopt interesting polygonal shapes. Both the cohesive and the strain energies vary inversely proportional to the square of the nanotube diameter. The armchair structure has larger strain energy than the zigzag structure, for a given diameter of SiCNTs. The calculated energy gap of SiCNTs decreases monotonically with increasing diameter and saturates at the calculated gap for graphitic sheet of SiC. The armchair structure has a larger energy gap than the zigzag structure, for a given diameter of SiCNTs. Theoretical predictions played an important role in guiding experimental studies in the past and we expect that many findings reported in this thesis may have important implications in the nanodevice applications of these interesting systems.