Rare earth elements (REEs) and actinides are critical to electronics, communication, military applications, and green energy systems. They also play a large role in nuclear waste challenges with critical national importance. Actinides are still among some of the least studied elements in the periodic table, due to their short half-lives and radioactivity, which demand expert facilities for research. Computational modeling greatly aids in understanding REEs and actinides; however, electronic structure modeling of these elements presents limitations. High Performance Computing (HPC) has had a direct impact not only on technical advances and access to information on a global scale but also on investigations of REEs and actinides. This work discusses recent advances in molecular and data driven modeling that are essential to the study of REEs and actinides, effects of computational science in nuclear and radiochemical applications, and advances and challenges in the exascale era of supercomputing.