Atomic nuclei are sensitive to the strong, weak, and electromagnetic interactions. In addition, nuclei are ideal laboratories to search for physics beyond the Standard Model, in processes such as neutrinoless double-beta decay, the direct detection of dark matter, or coherent neutrino-nucleus scattering. All these are governed by nuclear matrix elements (NMEs) that depend on the structure of the nuclei involved. Solving QCD for many-body systems at low energies, however, is beyond current capabilities. Alternatively, modern ab initio calculations can now solve the nuclear many-body problem with interactions derived from chiral effective field theory, an effective theory with the symmetries of QCD. In this seminar I will present ab initio results for the nuclear structure of heavy nuclei around nickel and tin. In addition, I will discuss NMEs for double-beta decay, and the scattering of neutrinos and WIMPs off nuclei, emphasizing the impact of NMEs in searches of new physics beyond the Standard Model.