The powerful technique of fitting a linear system model to the input–output response data with a least-squared error can be made even more useful by developing a recursive form with limited signal data memory to adapt with nonstationary systems and signals. This chapter shows that simplifications to the recursive least-squares (RLS) algorithm which require very few operations but converge more slowly. Approximations to RLS can offer significant computational savings, but at the expense of slower convergence to the least-squared error solution for the optimal filter. The least-mean-square (LMS) algorithm offers significant savings in computational complexity over the RLS algorithm for adaptive filtering. Signal modeling using adaptive signal-whitening filters is another basic adaptive filtering operation. The convergence properties of the LMS and RLS algorithms have been presented for two basic adaptive filtering tasks: system identification and signal modeling. The LMS algorithm offers significant savings in computational complexity over the RLS algorithm for adaptive filtering.