It is noticed that a considerable class of problems arising from systems and control and related fields may be reduced to parameter estimation, which, in turn, can be transformed to a root-seeking problem for unknown functions. The paper first introduces the root-seeking method based on the noisy observations, i.e., the classical stochastic approxi- mation algorithm. Against the restrictions of applying the classical algorithm, the stochastic approximation algorithm with expanding truncations (SAAWET) is introduced, and its general convergence theorem is demonstrated as well. Then, SAAWET is applied to solve problems like coefficient identification and order determination of linear stochastic systems, identification of Hammerstein, Wiener, and NARX systems, iterative learning control and adaptive regula- tion for nonlinear stochastic systems, and some others. All estimates given by the method are recursive and converge to the corresponding true values with probability one.