We review a series of results that have been obtained in the context of the DFG-SPP 1324 project “Adaptive wavelet methods for SPDEs”. This project has been concerned with the construction and analysis of adaptive wavelet methods for second order parabolic stochastic partial differential equations on bounded, possibly nonsmooth domains O ⊂ ℝ^d. A detailed regularity analysis for the solution process u in the scale of Besov spaces (Formula Presented), α > 0, p ≥ 2, is presented. The regularity in this scale is known to determine the order of convergence that can be achieved by adaptive wavelet algorithms and other nonlinear approximation schemes. As it turns out, in general, for solutions of SPDEs this regularity exceeds the L_p(O)-Sobolev regularity, which determines the order of convergence for uniform approximation schemes. We also study nonlinear wavelet approximation of elliptic boundary value problems on O with random righthand side. Such problems appear naturally when applying Rothe’s method to the parabolic stochastic equation. A general stochastic wavelet model for the righthand side is introduced and its Besov regularity as well as linear and nonlinear approximation is studied. The results are matched by computational experiments.