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A main problem when studying any mathematical property is to determine its stability, i.e., under what type of perturbations it is preserved. With this aim, here we study the stability of Gromov hyperbolicity, a property which has been proved to be fruitful in many fields. First of all we analyze the stability under appropriate limits, in the context of general metric spaces. We also prove the stability under some transformations in Riemann surfaces, even though the original surface and the modified one are not quasi-isometric.