Slope stability assessments are classical problems for geotechnical engineers. The predictions of slope stability in soil or rock masses play an important role when designing for dams, roads, tunnels, excavations, open pit mines and other engineering structures. Stability charts continue to be used by engineers as preliminary design tools and by educators for training purposes. However, the majority of the existing chart solutions assume the slope problem is semi-infinite (plane-strain) in length. It is commonly believed that this assumption is conservative for design, but non-conservative when a back-analysis is performed. In order to obtain a more economical design or more precise parameters from a back-analysis, it is therefore important to quantify three dimensional boundary effects on slope stability. A significant aim of this research is to look more closely at the effect of three dimensions when predicting slope stability. In engineering practice, the limit equilibrium method (LEM) is the most popular approach for estimating the slope stability. It is well known that the solution obtained from the limit equilibrium method is not rigorous, because neither static nor kinematic admissibility conditions are satisfied. In addition, assumptions are made regarding inter slice forces for a two dimensional case and inter-column forces for a three dimensional case in order to find a solution. Therefore, a number of more theoretically rigorous numerical methods have been used in this research when studying 2D and 3D slope problems. In this thesis, the results of a comprehensive numerical study into the failure mechanisms of soil and rock slopes are presented. Consideration is given to the wide range of parameters that influence slope stability. The aim of this research is to better understand slope failure mechanisms and to develop rigorous stability solutions that can be used by design engineers. The study is unique in that two distinctly different numerical methods have been used in tandem to determine the ultimate stability of slopes, namely the upper and lower bound theorems of limit analysis and the displacement finite element method. The limit equilibrium method is also employed for comparison purposes. A comparison of the results from each technique provides an opportunity to validate the findings and gives a rigorous evaluation of slope stability.