Modern data is often composed of two (or more) morphologically distinct constituents -- for instance, pointlike and curvelike structures in astronomical imaging of galaxies. Although it seems impossible to extract those components -- as there are two unknowns for every datum -- suggestive empirical results have already been obtained. In this talk we present a theoretical approach to the Geometric Separation Problem based on deliberately overcomplete systems which sparsify the different components and l1- minimization as the decomposition technique. Our analysis has two interesting features. Firstly, we use a viewpoint deriving from microlocal analysis to understand heuristically why separation might be possible and to organize a rigorous analysis. Secondly, our novel technical tools open a new direction for the recent avalanche of results on sparsity and l1- minimization by using the geometry of a problem as the driving force. As one application we prove that for images comprising pointlike and curvelike structures at all sufficiently fine scales, nearly-perfect separation is achieved when choosing radial wavelets and curvelets or orthonormal wavelets and shearlets as sparsifying systems. This is joint work with David Donoho (Stanford