Adaptive filtering algorithms are considered in this work. The main effort is devoted to improve the performance of such algorithms. Two classes of algorithms are given. The first one uses averaging in the approximation sequence obtained via slowly varying gains, and the second one utilizes averages in both the approximation sequence and the observed signals. Asymptotic properties--convergence and rate of convergence are developed. Analysis to one of the algorithms is presented. It is shown that the averaging approach gives rise to asymptotically optimal performance and results in asymptotically efficient procedures.