The global transition to low-carbon energy sources will require a significant contribution of nuclear energy to achieve emission goals. Low-level radioactive wastes (LLW) and intermediate-level radioactive wastes (ILW) are created in various phases of the nuclear fuel cycle for power generation, as well as from nuclear accidents, legacy weapons production, contaminated site decommissioning, and other nuclear activities such as radiopharmaceutical production. In this review, we will summarize recent developments, state-of-the-art glass formulations, and thermal treatment process developments for vitrification of nuclear LLW and ILW from programs in Europe, Asia, Australia, and North America. Throughout, we will discuss the selection of glass over other possible waste forms and any special processing considerations due to the nature of the waste. The characteristics of the wastes, such as mixed technological waste, waste coming from dismantling of reprocessing facilities, site decommissioning, and accident site decontamination, are important considerations. This is balanced with the suite of technologies available to vitrify these wastes, e.g., variations of incineration, in-can melting, and plasma treatment. Glass properties and microstructural aspects are compared to give an overview of the versatility of packaging matrices, such as homogeneous glasses, composites, and crystalline matrices. The volume and heterogeneity of the waste, specific radionuclide content, and solubility of components in silicate melts, all factor into the selection of a given waste form, processing route, and technology. Case studies include examples from the United States, United Kingdom, the Russian Federation, France, Australia, Japan, Korea, and China.