The present project aims at studying the mathematical theory of tensor approximation schemes for representing high-dimensional functions. Starting point are high-dimensional functions, which are defined on an m-fold product of domains of arbitrary dimensions, providing smoothness either in standard Sobolev spaces or in Sobolev spaces of dominating mixed smoothness. Of practical interest is also the situation that such spaces are equipped with dimension weights. We will provide estimates on the cost to approximate such functions in terms of the required ranks and also in terms of storage requirements for the eigenfunctions.