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We highlight some recent new developments concerning the sparse represen-tation of possibly high-dimensional functions exhibiting strong anisotropic features and low regularity in isotropic Sobolev or Besov scales. Specifically, we focus on the solution of transport equations which exhibit propagation of singularities where, additionally, high-dimensionality enters when the convection field, and hence the solutions, depend on parameters varying over some compact set. Important con-stituents of our approach are directionally adaptive discretization concepts moti-vated by compactly supported shearlet systems, and well-conditioned stable varia-tional formulations that support trial spaces with anisotropic refinements with arbi-trary directionalities. We prove that they provide tight error-residual relations which are used to contrive rigorously founded adaptive refinement schemes which con-verge in L2. Moreover, in the context of parameter dependent problems we discuss two approaches serving different purposes and working under different regularity assumptions. For “frequent query problems”, making essential use of the novel well-conditioned variational formulations, a new Reduced Basis Method is outlined which exhibits a certain rate-optimal performance for indefinite, unsymmetric or singularly perturbed problems. For the radiative transfer problem with scattering a sparse tensor method is presented which mitigates or even overcomes the curse of dimensionality under suitable (so far still isotropic) regularity assumptions. Numer-ical examples for both methods illustrate the theoretical findings