This thesis is concerned with the development of ab-initio molecular dynamics (AIMD) using density functional theory, and its application to simple diatomics and small sodium clusters. The approach follows Car and Parrinello and the local spin density (LSD) approximation is taken as a starting point for the description of the electronic states in these systems. An improvement to LSD by correcting for the spurious self-interaction terms (SIC), as proposed by Perdew and Zunger, and the way in which this can be implemented is considered. The SIC corrected LSD is tested on simple diatomic molecules, and is shown, especially for the case of H₂, to significantly improve on the LSD description of the potential energy curve and the spin pairing/unpairing transition. LSD and SIC are then compared for small sodium clusters, for which the geometries, binding energies, and polarizabilities are determined, as well as vibrational frequencies for two clusters. Conclusions are hampered by the lack of experimental data, and uncertainties due to the pseudopotential. However SIC gives a more accurate description of the binding energies, and the unique SIC orbitals allow a description of the bonding which accurately predicts the relative stabilities of the clusters and rationalises their geometries. The problems of breakdown of the Car-Parrinello (CP) method due to "nonadiabatic" effects is considered and it is shown that the problems arising during bond formation and bond breaking, as illustrated by the case of Na₂, can largely be overcome by applying an external thermostat to the fake degrees of freedom. This method is then tested using the dynamics of small sodium clusters. In particular the dynamics of the pseudorotation of Na₃ is studied and it is shown that the mechanism and information about the kinetics can be determined. Finally other methods for improving upon LSD are considered. It is shown that gradient corrections using a plane wave basis set have inherent problems in implementation. Implementation of the exact Hartree-Fock exchange is considered, and this in conjunction with a form of self-interaction correction on the correlation is shown give very accurate results for simple diatomics which are cheaper than using full configuration interaction methods.