Stability theorems for an adaptive iterative learning control (ILC) system are motivated and described in compact terms that relate the properties of the schemes to systems structure and other important aspects of systems dynamics. The use of high gain feedback is reviewed and a full proof of the convergence of a 'Universal' adaptive scheme based on adaptive gain concepts is given. The results indicate clearly that successful ILC can be achieved in the presence of substantial uncertainty in the detailed knowledge of plant parameters and order. They also suggest that the form and success of the controller will depend crucially on the plant's relative degree and also on its minimum-phase properties.