The purpose of stochastic approximation is to find the roots of an unknown function f(·), which can be observed, but the observations are corrupted by errors. General convergence theorems for stochastic approximation algorithms with expanding truncations are presented. The observation errors are allowed to include both random noise and structural uncertainties. The conditions imposed on the observation errors are the weakest possible, while the function f(·) is only required to be measurable and locally bounded. Applications of the general convergence theorems to optimization and signal processing demonstrate the strong points of results given in the paper.