New fixed-order fast transversal filter (FTF) algorithms are introduced for several common windowed recursive-least-squares (RLS) adaptive-filtering criteria. O(N) operations per data point, where N is the filter order, are required by the new algorithms. These algorithms are characterized by two different time-variant scaling techniques that are applied to the internal quantities, leading to normalized and over-normalized FTF algorithms. It is this scaling that distinguishes the new algorithms from the multitude of fast-RLS-Kalman or fast-RLS-Kalman-type algorithms that have appeared in the literature for these same windowed RLS criteria, and which use no normalization or scaling of the internal algorithmic quantities. The overnormalized fast transversal filters have the lowest possible computational requirements for any of the considered windows. The normalized FTF algorithms are then introduced, at a modest increase in computational requirements, to significantly mitigate the numerical deficiencies inherent in all most-efficient RLS solutions, thus illustrating an interesting and important tradeoff between the growth rate of numerical errors and computational requirements for all fixed-order algorithms. Performance of the algorithms, as well as some illustrative tracking comparisons for the various windows, is verified via simulation.