Azines are important in many extraterrestrial environments, from the atmosphere of Titan to the interstellar medium. They have been implicated as possible carriers of the diffuse interstellar bands in astronomy, indicating their persistence in interstellar space. Most importantly, they constitute the basic building blocks of DNA and RNA, so their chemical reactivity in these environments has significant astrobiological implications. In addition, N and O atoms are widely observed in the ISM and in the ionospheres of planets and moons. However, the chemical reactions of molecular anions with abundant interstellar and atmospheric atomic species are largely unexplored. In this paper, gas-phase reactions of deprotonated anions of benzene, pyridine, pyridazine, pyrimidine, pyrazine, and s-triazine with N and O atoms are studied both experimentally and computationally. In all cases, the major reaction channel is associative electron detachment; these reactions are particularly important since they control the balance between negative ions and free electron densities. The reactions of the azine anions with N atoms exhibit larger rate constants than reactions of corresponding chain anions. The reactions of azine anions with O atoms are even more rapid, with complex product patterns for different reactants. The mechanisms are studied theoretically by employing density functional theory; spin conversion is found to be important in determining some product distributions. The rich gas-phase chemistry observed in this work provides a better understanding of ion-atom reactions and their contributions to ionospheric chemistry as well as the chemical processing that occurs in the boundary layers between diffuse and dense interstellar clouds.