This paper considers identification of Wiener systems for which the intal variables and output are corrupted by noises. When the intal noise is a sequence of independent and identically distributed (iid) Gaussian random variables, by the Weierstrass transformation (WT) the system under consideration ts to be a Wiener system without intal noise. The nonlinear part of the latter is nothing else than the WT of the nonlinear function of the original system, while the linear subsystem is the same for both systems before and after WT. Under reasonable conditions, the recursive identification algorithms are proposed for the transformed Wiener system, and strong consistency for the estimates is established. By using the inverse WT the nonparametric estimates for the nonlinearity of the original system are derived, and they are strongly consistent if the nonlinearity in the original system is a polynomial. Similar results also hold in the case where the intal noise is non-Gaussian. Simulation results are fully consistent with the theoretical analysis.