Several of the most promising searches for physics beyond the Standard Model are led by low-energy experiments. Typically, these searches involve atoms and the sensitivity to new physics is given by interactions involving the atomic nucleus. Therefore, the structure of these nuclei is key to interpret the experimental results. This information is usually encoded into a nuclear matrix element. In this seminar, I will focus on searches for neutrinoless double-beta decay, electric dipole moments, and the direct detection of dark matter. I will discuss recent progress in calculations of the nuclear matrix elements, and emphasize the connection between them and the nuclear structure of the nuclei involved.