Over the last decade, considerable effort has been devoted to the implementation and analysis of self-tuning controllers on systems which are assumed to be represented exactly by linear dynamical models. In this thesis we examine the robustness of the self-tuning controller, when applied to systems consisting of a nominal linear plant which may have linear or nonlinear perturbations. Robust stability is the primary criterion and most of the results are for the Clarke-Gawthrop version of the self-tuning controller. Conditions are derived for the robust stability of the adaptively controlled system in terms of the design choices available to the engineer setting up the self-tuning controller. These are strong stability results in that they are in terms of both 1₂ and 1_∞ stability. The results are shown to be applicable to the general delay case and in the presence of non-zero mean disturbances. Preliminary results are also obtained for the robust stability of the explicit self-tuning controller.