Increasing demand for clean, globally available energy, provokes the development of alternative or nonconventional energy storage sources with higher energy density and power delivery. Therefore, supercapacitors have received great attention in both academic and industrial research. In addition, supercapacitors are promising new sources of energy in the future energy technology. Hence, rapid progress has been made to understand their fundamentals and applicable aspects. Supercapacitors, also known as ultracapacitors, bridge the gap between batteries and con- ventional capacitors, which means that supercapacitors can accept and deliver charge much faster than batteries, and can store 10 to 100 times more energy per unit volume than fuel cells. Su- percapacitors are well known not only because of their huge energy density, but also due to their long shelf and good cyclic ability. The particular properties of supercapacitors are due to utiliza- tion of electrode materials with very high porosity, and also the specific mechanism of charge storage. In general, there are two fundamental chemical and physical mechanisms for energy storage. In chemical mechanism the charges are released through oxidation-reduction reaction. However, in the physical mechanism the electrical energy is stored physically with electrostatic interaction, while no chemical and phase changes occur. Hence, according to theses charge storage mechanisms supercapacitors are divided into three main groups: electric double layer capacitors (EDLCs), pseudocapacitors and hybrid capacitors. Each of these supercapacitors has its own ad- vantages and disadvantages. These three different kind of supercacaitors are distinguished by their charge storage mechanism and also their electrode materials. Electrical double layer capacitors store the charge based on physical mechanism, and also various types of carbon are used as electrode material. Pseudocapacitors save energy via electro- chemical redox reactions. Besides, these capacitors utilize metal oxides and conducting polymers as electrode material. Hybrid capacitors are another class of supercapacitors, which are constructed of two different types of electrode materials. Therefore, the mechanism of energy storage is a com- bination of both chemical and physical mechanisms. The most used electrode materials in these supercapacitors are a composition of carbon-based materials with either conducting polymers or metal oxide materials. In this work electrical double layer capacitors have captured our attention because of both their interesting energy storage mechanism and also their electrode materials. Electrode material is one of the most important factors in the performance of an electro- chemical energy storage device. Therefore, innovation of new electrode materials is one of the most attractive topics in recent investigations. Especially carbon based electrodes such as carbon nanofibers, activated carbon, carbon nanotubes (CNTs) etc. with their porous structure can pro- vide very huge surface area, consequently immense capacity. Among the examined carbon based materials carbon nanotubes are one of the most interesting because of their unique physical and chemical properties. In fact, special properties of carbon nanotubes such as individual tubular structure, very high chemical stability, low resistivity, high thermal and electrical conductivity and enormous surface area make them good candidates for electrode material in electrical double layer capacitors. Carbon nanotubes are divided into three main groups such as zigzag, armchair and chi- ral tubes. Based on their electronic structure, carbon nanotubes can have metallic or semimetallic characteristic. As mentioned above the huge capacity of carbon materials is due to their massive surface area. Furthermore, in the case of carbon nanotubes both the inner and outer walls can be available for electrolyte ions. At first it was thought, that very narrow pores do not participate in the forma- tion of double layer and energy storage. However, experimental investigations proved, that very narrow pores (lower than 1nm size) not only participate in energy storage, but also exhibit an enor- mous increase of capacitance. Later, theoretical findings showed that the image charge between ion and pore wall screen the repulsion between the ions. This leads to a denser packing of ions, consequently increasing the electrode capacitance. In this work we have studied ion intercalation into carbon nanotubes with diameters lower than 1nm as electrode material by density functional theory. All the calculations have been done using the VASP package. The idea of this work is in that we have imagined carbon nanotubes as electrode materials immersed in solution. Hence, the electrolyte ions try to penetrate into the carbon nanotubes. This work is divided into four parts, which are as follows: In the first part, we have selected truncated carbon nanotubes or carbon nanorings, whoese ends are saturated with hydrogen atoms. In particular, the truncated carbon nanotubes are include the (6,0), (8,0), (10,0) and (12,0) carbon nanotubes. As electrolyte ions alkali (Li, Na and Cs) and halogen (Cl, Br and I) atoms have been tried. Meanwhile, we have neglected the presence of solvent or any counterpart. After simulation it was realized that all the alkali atoms have lost one electron, and also the halogen atoms have obtained one more electron, and also the stable position of all the ions is in the center of the tubes. The results have shown that the surrounding tubes screen the ionic charge very effectively, thereby the ion-ion interactions are strongly reduced, which explains, why narrow tubes store charge more effectively than wider one. We have calculated the insertion energies of the atoms into the tubes and understood that for each atom the diameter of the tube has to be optimized. In the second part of the work, we extended the model of short nanotubes to infinite ones. In particular, we have chosen the (6,2)CNT, the (6,3)CNT, the (8,0)CNT and the (5,5)CNT as electrode material. Among the presented carbon nanotubes the (6,2)CNT and the (8,0)CNT are semimetallic, while the (5,5)CNT and the (6,3)CNT are metallic. Like in the previous work we have inserted alkali (Li, Na and Cs) and halogen (F, Cl, Br and I) atoms into the carbon nanotubes. The results have shown that the atoms were fully ionized. The charge exchange with the CNTs affects the band structure, and turns those tubes that were originally semiconductors into conductors. None of the ions is adsorbed chemically, their position inside the tube and their energies of adsorption are determined by a competition between electrostatic image interactions, which favor a position at the wall, and Pauli repulsion. In models for charge storage it is often assumed that in small tubes the ions are at the center, but we have found several cases where small alkali ions are positioned near the wall. We have also investigated the screening of the Coulomb potential along the axis of the tubes. In particular we wanted to see if there is a difference between semiconducting and conducting CNTs. Within the accuracy of our calculations we found no difference in the screening, because the charge transfer has made the non-chiral tubes conducting. In the third part of this work, we have investigated insertion of alkali and halogen atoms into nitrogen doped (N-doped) carbon nanotubes. Here, the (8,0)CNT and the (5,5)CNT have been chosen as electrode material. The results have shown that N-doped carbon nanotubes are less stable than the pure ones, and also the atoms were fully ionized. The position of the ions in the carbon nanotubes exhibits contradictory behavior in the (8,0)CNT than the (5,5)CNT. In fact, in the former one the ions have high repulsion from the impurity area and try to get away as far as possible. This effect is stronger in the case of small ions like Li+. However, in the (5,5)CNT the ions get closer to the nitrogen area. This behavior is caused by the difference in the spin density of carbon nanotubes. In other words, the spin density is more localized in the N-doped (8,0)CNT than the N- doped (5,5)CNT. Therefore, it causes big repulsion with the inserted ions. The nitrogen doping and charge exchange with ions affect the band structure of carbon nanotubes. Actually, substitution of nitrogen and insertion of alkali atom keep the tubes conducting. However, intercalation of halogen atoms causes both tubes to become semiconducting. We also have calculated insertion energy of ions in the N-doped carbon nanotubes. The results have shown that insertion of the ions is more favorable in the N-doped carbon nanotubes than in the pure ones. In the fourth part of the work, we have taken into account the presence of counterions. For this purpose we have tried insertion of alkali halide monomers (LiF, LiCl and NaCl), a chain of NaCl in nanotube, as well as presence of water molecules with NaCl monomer in a carbon nanotube. Therefore, the (5,5)CNT as electrode material has been chosen. In all the investigated systems, the ions and the molecules are not chemically bound to the carbon nanotube wall. In the case of alkali halide monomers there is no strong charge transfer with the nanotube. However, for the chain of NaCl this interaction is stronger, while the main charge transfer is between the alkali and halide ions in the chain. Water does not exchange charge with the nanotube. Nonetheless, it tries to hydrate the alkali and halide ions confined in the tube.