In this talk, we give an overview of adaptive wavelet methods for solving operator equations. In particular, we will focus on the following topics: The application of these methods to time evolution problems as parabolic problems and the instationary Stokes system; the advantage of the application of tensor product wavelets and the role of anisotropic regularity; the construction of piecewise tensor product wavelet bases on general domains; the application of the adaptive scheme to singularly perturbed problems.