Nonstationary time series arise in many different disciplines, and there are many different reasons for wishing to study them. The particular interest in this thesis is in modelling the time series so as to obtain certain parameters of interest from it. Whatever the reason for studying such a time series and whatever the method chosen, in order to accommodate the nonstationarity of the series it is important to use an adaptive algorithm whose parameters are permitted to vary with time. The first aim of this thesis will be to examine existing adaptive algorithms, highlighting their strengths and weaknesses to determine which, if any, offers the best way forward towards developing new algorithms. Following this, rather than consider a specific class of algorithm a generic algorithm which contains the properties of more than one class of algorithm will be examined. To facilitate the development of this algorithm hyperparameters and hypermodels will be introduced. Results of simulations run to test the algorithms performance will be given. The second aim of this thesis will be to develop a new algorithm, the fast adaptive forward backward least squares algorithm. This algorithm incorporates a 'forgetting factor' to enable the tracking of nonstationary signals. Simulations will be performed which show that the algorithm can outperform the unwindowed version in the presence of a nonstationary signal. Stabilization techniques will be introduced which will prevent the algorithm exhibiting numerical instabilities to which this type of algorithm are prone. Simulations results will be presented to give guidelines for the choice of values of feedback gains which are to be used to prevent the exhibition of instability. Finally the advantages and limitations of both the new and existing algorithms will be summarized and suggested areas of future research outlined.