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The authors propose an advanced synthetic aperture radar (SAR) image formation framework based on iterative inversion algorithms that approximately solve a regularised least squares problem. The framework provides improved image reconstructions, compared to the standard methods, in certain imaging scenarios, for example when the SAR data are under-sampled. Iterative algorithms also allow prior information to be used to solve additional problems such as the correction of unknown phase errors in the SAR data. However, for an iterative inversion framework to be feasible, fast algorithms for the generative model and its adjoint must be available. The authors demonstrate how fast, N2 log2N complexity, (re/back)-projection algorithms can be used as accurate approximations for the generative model and its adjoint, without the limiting geometric approximations of other N2 log2N methods, for example, the polar format algorithm. Experimental results demonstrate the effectiveness of their framework using publicly available SAR datasets.