This work describes the applications of ab-initio quantum chemical methods to the studies of atomic clusters. In Chapter 1 a general description of quantum chemical methods, used to solve the stationary Schrodinger equation in subsequent parts of the dissertation, is given. In Chapter 2 the adsorption of oxygen molecules on small neutral and anionic gold clusters is studied. It is shown that O2 binds better to clusters with an odd number of electrons than to clusters with an even number of electrons. DFT results are found to be in significant disagreement with high-level ab-initio CCSD(T) results. Chapter 3 describes the study of reaction mechanisms of molecular hydrogen with small neutral and anionic gold clusters. The binding energies of one and two H2 molecules are calculated. The transition states of H2 dissociation on gold clusters are located. In contrast to O2 absorption, DFT produces reasonable results for the H2 binding energies and the barriers to H2 dissociation. The study of the stability of different isomers of C36 carbon clusters is presented in Chapter 4. It is shown that the singlet state of the lowest energy D6h isomer has significant diradical character. The experimental data is explained based on multireference perturbation theory calculations. It is shown that strong electron correlation is responsible for the high stability of D6h isomer observed in experiments. In Chapter 5 the mixed metal-carbon Ti8C12 cluster is studied with the main goal to determine the geometry and ground electronic state of this cluster. It is shown that the Td structure with a 1E ground state is a subject to Jahn-Teller distortion. The distorted D2d and C3v structures are studied with multireference configuration interaction and coupled cluster methods. The D2d structure with a singlet ground state suggested as a ground state of the Ti8C12 cluster. A new approach for solving the many electron Schrodinger equation is proposed in Chapter 6. In contrast to the wave function or the density functional theory approaches, the proposed method uses the first-order reduced density matrix and the diagonal part (density) of the cumulant of the second-order reduced density matrix.