This article deals primarily with the selection criteria, production, and the nuclear, physical, and chemical properties of certain dual-purpose radionuclides, including those that are currently being used, or studied and evaluated, and those that warrant future investigations. Various scientific and practical issues related to the production and availability of these radionuclides is briefly addressed. At brookhaven national laboratory (BNL), we have developed a paradigm that involves specific individual 'dual-purpose' radionuclides or radionuclide pairs with emissions suitable for both imaging and therapy, and which when molecularly (selectively) targeted using appropriate carriers, would allow pre-therapy low-dose imaging plus higher-dose therapy in the same patient. We have made an attempt to sort out and organize a number of such theragnostic radionuclides and radionuclide pairs that may thus potentially bring us closer to the age-long dream of personalized medicine for performing tailored low-dose molecular imaging (SPECT/CT or PET/CT) to provide the necessary pretherapy information on biodistribution, dosimetry, the limiting or critical organ or tissue, and the maximum tolerated dose (MTD), etc., followed by performing higher-dose targeted molecular therapy in the same patient with the same radiopharmaceutical. As an example, our preclinical and clinical studies with the theragnostic radionuclide Sn-117m are covered in somewhat greater detail.A troublesome problem that remains yet to be fully resolved is the lack of availability, in sufficient quantities and at reasonable cost, of a majority of the best candidate theragnostic radionuclides in a no-carrier-added (NCA) form. In this regard, a summary description of recently developed new or modified methods at BNL for the production of five theragnostic radionuclide/radionuclide pair items, whose nuclear, physical, and chemical characteristics seem to show promise for therapeutic oncology and for treating other disorders that respond to radionuclide therapy, is provided.