The paper is dedicated to a very interesting and rapidly developing field of nuclear physics—the generation and study of exotic nuclei in the vicinity of the driplines. The history of this field is presented with methods of obtaining such nuclei in the accelerators of the world’s leading research centers—the European Organization for Nuclear Research (CERN) in Switzerland and the Helmholtz Center for Heavy Ion Research (GSI) in Germany. The structure of the nuclei, as they change greatly approaching the driplines of neutron and proton stability, is given, as well as the results of experimental research of neutron- and proton-rich nuclei and the formation of neutron halos in isotopes of helium, lithium, beryllium, and boron, strongly enriched with neutrons. Information on medical applications of radionuclide beams is presented.