We propose a numerical method to simulate and invert the two-dimensional attenuated Radon transform (AtRT) from full (360°) or partial (180°) measurements. The method is based on an extension of the fast slant stack algorithm developed for the Radon transform. We show that the algorithm offers robust and fast inversion of the AtRT for a wide class of synthetic sources and absorptions. The complexity of the fast algorithm to compute the AtRT of a n × n image and perform the reconstruction from the AtRT data is O(Nn²log n) operations, with N the number of Fourier modes necessary to accurately represent the absorption map. The algorithm is applied to the reconstruction of the exponential Radon transform, where the absorption coefficient is constant, and of the AtRT when only 180° measurements are available. The reconstruction from partial measurements is based on an iterative scheme introduced recently in Bal (2004 Inverse Problems 20 399–419). Single-photon emission computed tomography is an important medical imaging technique based on the inversion of the AtRT.