Abstract: Several methods have been recently presented in the literature for performing very efficiently l1-norm and total variation minimization, e.g., for image processing problems of small or medium size. However, because of their iterative-sequential formulation, none of the mentioned methods is able to address in real-time, or at least in an acceptable computational time, extremely large problems, such as 4D imaging (spatial plus temporal dimensions) from functional magnetic-resonance in nuclear medical imaging, astronomical imaging or global terrestrial seismic tomography. For such large scale simulations we need to address methods which allow us to reduce the problem to a finite sequence of sub-problems of a more manageable size, perhaps computable by one of the methods above. With this aim we present in this talk subspace correction and domain decomposition methods both for l1-norm and total variation minimizations. We review sequential and parallel domain decomposition strategies for solving l1-minimization. These methods proved to be particularly efficient, for instance, in large scale denoising problems. Then we address the complicated problem of formulating efficient overlapping and nonoverlapping domain decomposition methods for total variation minimization. These are the first successful attempts of addressing such a strategy for the nonlinear, nonadditive, and nonsmooth problem of total variation minimization. In particular we are able to present a sketch of their convergence proof. We show also several convincing numerical experiments, with successful applications of the algorithms for the restoration 1D and 2D signals in interpolation/inpainting problems respectively, and in a compressed sensing problem, for recovering piecewise constant medical-type images from partial Fourier ensembles. We conclude by mentioning our work in progress on multiscale methods based on wavelet space decompositions for total variation minimization. We show our preliminary and very promising numerical results.