Nonlinear approximation and function space of dominating mixed smoothness As the main object of this thesis we study a generalization of function spaces with dominating smoothness. The basic idea for this generalization is based on a splitting of the d variables into N groups of possibly varying length. In this way the usual spaces of dominating mixed smoothness as well as the classical isotropic spaces occur as special cases (for d groups of length one each or for 1 group of length d respectively). More precisely we will study first Sobolev type spaces and afterwards Besov and Triebel Lizorkin type spaces are introduced. For these spaces several basic properties such as Fourier multipliers and duality are discussed. As the main tool for further studies characterizations by local means are proved. The main result of the first part of the thesis consists in a characterization by tensor product Daubechies wavelets. One immediate corollary of this characterization is the identification of certain Besov spaces of dominating mixed smoothness as tensor products of isotropic ones establishing a connection to many recent discussions in high dimensional approximation.