We consider the inversion of the two-dimensional attenuated Radon transform (AtRT) from full or partial measurements. The AtRT is routinely inverted in SPECT (Single Photon Emission Computed Tomography), a popular medical imaging technique. We show that two spatially independent source terms can be reconstructed from the AtRT. This is based on an extension of the recent Novikov formula and on recasting the inversion as a Riemann Hilbert problem. Next we consider the reconstruction of one spatially dependent source term from half of the angular measurements ($180^\circ$ measurements). We show that under a smallness condition on the gradient of the known absorption map, compactly supported source terms can uniquely be reconstructed. An iterative procedure is presented. Finally we consider a fast, robust, and accurate technique to compute and invert the AtRT. The numerical technique is based on a generalization of the fast slant stack algorithm, which performs very well to compute and invert the classical Radon transform. The technique is very accurate for moderate values of the absorption map. Modifications are proposed in the case of larger absorption maps. Numerical simulations complement the theory and show the robustness of the method.