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In this talk, we discuss some of the recent developments in sparsity promoting optimization methods for sensor array imaging which is an important technique with applications in radar and sonar imaging, seismic imaging and medical imaging. There are various conditions on the unknown which are known to be sufficient for these methods to achieve an exact reconstruction. However, these conditions do not hold in many applications. We present a theory that quantifies the quality of the reconstruction when these conditions fail and also helps us understand thecases where additional measurements can improve the reconstruction. As another application of this theory, we will discuss its possible usage in reweighted l1 minimization algorithms. No background other than familiarity with linear algebra will be assumed for this talk, and it will be accessible to graduate students.