Joint with R. Hiptmair, Konstantin Grella, Eividn Fonn of SAM, ETH. We report on an ongoing project on Sparse Tensor Finite Element Discretizations for High Dimensional Linear Transport Problems. After reviewing several well-posed variational formulations and the regularity of weak solutions of these problems, we discuss their stable discretizations, with a focus on hierarchic, multilevel type discretizations. Particular examples include (multi)wavelet and shearlet discretizations. We discuss sparse tensor discretizations for Least Squares Formulations of first order transport equations on high dimensional parameter spaces. The formulation is due to Manteuffel etal. (SINUM2000). We present preliminary numerical results for both, sparse tensor spectral as well as for wavelet discretizations. Results are report from ongoing work at the Seminar for Applied Mathematics at ETH Zurich which is supported by the Swiss National Science Foundation (SNF) and from joint work with the groups of W. Dahmen and of G. Kutyniok within the Priority Research Programme (SPP) No. 1324 of the German Research Foundation. http://www.dfg-spp1324.de