The development of the Standard Model (SM) has been one of the crowning achievements in modern physics, and is the cornerstone of current subatomic studies. Despite its success, the SM is known to be incomplete, and providing limits on possible physics beyond the Standard Model (BSM) is crucial to our understanding of the natural universe. Although they are generally complex, atomic nuclei can be exploited as a laboratory for these studies through the use of rare-isotope beams (RIBs). The production of these short-lived, very exotic isotopes has opened new avenues of research in our search for BSM physics in the era of the LHC. This work is at the precision and sensitivity frontiers, and helps to bridge the gap between atomic, nuclear, and particle physics using novel, state-of-the-art detection techniques. In this talk, I will use these topics to highlight the significant role of the atomic nucleus in our ongoing search for additional generations of quarks, new descriptions of the weak interaction, and light dark matter. These studies play a critical role in providing the groundwork for our quest to develop the "New Standard Model".