A unified and consistent formulation is developed for both filtered reference and filtered error forms of the instantaneous steepest descent, or LMS, algorithm when used to adapt FIR feedforward controllers. Both algorithms minimise the mean-square value of the same, output, error function. The two algorithms are first formulated for single-input single-output linear plants. It is argued that since the behaviour of the two algorithms is equivalent in the case of slow adaptation, the conditions on the accuracy of the plant model for stability should also be the same in both cases. This is expressed as a bound on the unstructured multiplicative uncertainty of the plant. Filtered reference and filtered error algorithms are also derived for multiple-input multiple-output (MIMO) linear systems, although the filtered reference algorithm is found not to have a simple block diagram interpretation. In the MIMO case, the filtered error form of the algorithm can have considerable computational advantages over the filtered reference form. Finally the two algorithms are extended to the case of non-linear plants and/or controllers which are modelled as feedforward neural networks. In the non-linear case the two formulations of the LMS algorithm reduce to two forms of the widely used backpropagation algorithm.