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Perovskite solar devices are nowadays the fastest advancing photovoltaic technology. Their large-scale application is however restrained by instability and toxicity issues. Alloying is a promising way to stabilize perovskites, optimizing at the same time their absorption and charge-transport properties. We perform an extensive computational study of the thermodynamic stability and electronic properties of oxynitride, oxyfluoride and nitrofluoride perovskites. We consider quaternary stoichiometries of the type ABX2Y, where A and B are any elements of the periodic table and X and Y are nitrogen, oxygen, or fluorine. As a starting point we explore the composition space using a simple five-atom perovskite unit cell. We then filter the candidate compositions according to their distance to the convex hull of thermodynamic stability. For the most stable systems, we then investigate other prototype structures, including more complex perovskite phases that allow for octahedral distortions, and a few non-perovskite geometries. Furthermore, for some paradigmatic cases, we study the effect of disorder by exhaustive enumeration of all possible disordered stoichiometric phases with up to 20 atoms in the unit cell. Our calculations are in very good agreement with data for experimentally known mixed anionic compounds, and predict a series of novel stable (perovskite and non-perovskite) oxynitride and oxyfluoride phases, including some with unexpected chemical composition, and one single nitrofluoride compound. Finally, we calculate and discuss the electronic properties of these compounds and their potential for application as photovoltaic absorbers.