The basic adaptive filtering algorithmX_{n+1}^{\epsilon} = X_{n}^{\epsilon} - \epsilon Y_{n}(Y_{n}^{'}X_{n}^{\epsilon} - \psi_{n})is analyzed using the theory of weak convergence. Apart from some very special cases, the analysis is hard when done for each fixed\epsilon > 0. But the weak convergence techniques are set up to provide much information for small\epsilon. The relevant facts from the theory are given. Definex^{\epsilon}(\cdot)byx^{\epsilon}(t) = X_{n}^{\epsilon}on[n\epsilon, n\epsilon + \epsilon). Then weak (distributional) convergence of\{x^{\epsilon}(\cdot)\}and of\{x^{\epsilon}(\cdot + t_{\epsilon})\}is proved under very weak assumptions, wheret_{\epsilon} \rightarrow \inftyas\epsilon \rightarrow 0. The normalized errors\{(X_{n}^{\epsilon} - \theta ) / \sqrt{\epsilon} \}are analyzed, where\thetais a "stable" point for the "mean" algorithm. The asymptotic properties of a projection algorithm are developed, where theX_{n}^{\epsilon}are truncated at each iteration, if they fall outside of a given set.