: Climate change necessitates urgent actions to mitigate carbon dioxide (CO2) emissions from fossil fuel-based energy generation. Among various strategies, the deployment of carbon capture and storage (CCS) solutions are critical for reducing emissions from point sources such as power plants and heavy industries. In this context, cryogenic carbon capture (CCC) via desublimation has emerged as a promising technology. While CCC offers high separation efficiency, minimal downstream compression work, and integration potential with existing industrial processes, challenges such as low operating temperatures and equipment costs persist. Ongoing research aims to address these hurdles in order to optimize the desublimation processes for widespread implementation. This review consolidates diverse literature works, providing insights into the strengths and limitations of CCC technology, including the latest pilot plant scale demonstrations. The transformative potential of CCC is first assessed on a theoretical basis, such as thermodynamic aspects and mass transfer phenomena. Then, recent advancements in the proposed process configurations are critically assessed and compared through key performance indicators. Furthermore, future research directions for this technology are clearly highlighted.