This thesis reports an investigation of density functional based methods for the study of selected organic reactions and simple atomic and molecular systems to evaluate and rationalize the role of exact (Hartree-Fock) exchange in hybrid density functional methods and the performance of density functional methods in the prediction of chemical property. The performance of density functional methods has been compared with certain conventional 'ab initio' methods: HF, MP2 and CCSD, and available experimental results. The chemical systems and density functional methods investigated were: '1. The reactions of a triplet oxygen atom' (' 3P') 'with C60 fragments.' The structures of C60 fragments and their oxides were studied at the HF, BHandHLYP, and B3LYP levels with the 6-31G* basis set. The reaction barrier and the transition state structure for the initial reaction of a triplet oxygen atom with these fragments were investigated theoretically. The nucleus-independent chemical shifts (NICSs) of Schleyer were adopted as criteria of aromaticity in polycyclic aromatic hydrocarbons (PAHs) and used to study the reactivity and properties of the C60 fragments. The initial reaction of a triplet oxygen atom with C60 fragments is exothermic with a low reaction barrier. '2. Reaction of triplet oxygen atom' (' 3P') 'with benzene and isomerization of benzene oxides. ' The reaction of a triplet oxygen atom and the potential energy surfaces for isomerization of the benzene oxides were studied to explore the reaction mechanism and find the reaction pathways. For these reactions, the role of exact exchange (Hartree-Fock (HF) exchange) was examined through the performances of the HF, BHandHLYP, B3LYP, B1LYP, BLYP and mPW1PW91 methods in the prediction of the reaction barrier and in potential energy surface modeling in comparison to the results from MP2. The 6-31G* basis set was used throughout. The theoretical predictions were compared with available experimental results when possible. '3. Numerical examination of density functional methods.' The exchange functionals B, PW91, mPW91, LG, G96 and the correlation functionals LYP and PW91 were examined numerically in combination with HF exchange to scrutinize the role of HF exchange in hybrid density functional theory. Both energy and electronic properties were studied for the simple atomic systems H, He and Li. One-electron systems (H, H2+, He +) were employed to investigate the self-interaction error in the exchange and correlation functionals. The highest occupied orbital energies of these systems were examined with the various exchange correlation functionals. A new combination of exchange and correlation functionals within the one-parameter protocol of the hybrid method, mPW1LYP, was evaluated on the reduced G2 data for geometric and thermodynamic predictions. This new method was employed in studies on some inert gas dimers, weakly interacting complexes, low barrier reactions, and the electronic spectra of some small molecules. The overall performance of mPW1LYP is promising.