By means of white noise analysis, we prove some limit theorems for nonlinear functionals of a given Volterra process. In particular, our results apply to fractional Brownian motion (fBm), and should be compared with the classical convergence results of the eighties by Breuer, Dobrushin, Giraitis, Major, Surgailis and Taqqu, as well as the recent advances concerning the construction of a Lévy area for fBm by Coutin, Qian and Unterberger