Beams of short-lived radioactive nuclei are needed for frontier experimental research in nuclear structure, reactions, and astrophysics. Negatively charged radioactive ion beams have unique advantages and allow for the use of a tandem accelerator for post-acceleration, which can provide the highest beam quality and continuously variable energies. Negative ion beams can be obtained with high intensity and some unique beam purification techniques based on differences in electronegativity and chemical reactivity can be used to provide beams with high purity. This article describes the production of negative radioactive ion beams at the former holifield radioactive ion beam facility at Oak Ridge National Laboratory and at the CERN ISOLDE facility with emphasis on the development of the negative ion sources employed at these two facilities.