Molecules and atoms can be used for precision measurements of the electromagnetic moments of nuclei, their charge radii, and manifestations of discrete-symmetry violation in fundamental interactions. Addressing all these problems requires high-accuracy atomic and molecular theory for the interpretation of measurements. The adaptation of modern relativistic coupled cluster methods [1] for calculating shielding constants in nuclear magnetic resonance problems makes it possible to extract the magnetic moments of stable nuclei from experimental data at a new level of accuracy and to resolve long-standing discrepancies between experiments and theoretical predictions for highly charged ions. Similar theoretical approaches can be applied to estimate the effect of P-odd interactions induced by the nuclear anapole moment on the nuclear spin-spin coupling constants in molecules [2]. A precision treatment of the hyperfine structure of atoms and molecules requires accounting for the effect of the nuclear magnetization distribution, which may be poorly known. The contribution of this effect to HFS can be factorized into a purely electronic factor and a universal parameter that does not depend on the electronic state of the atom or molecule and determines the magnetization distribution [3]. This method makes it possible to avoid the need to know this distribution and enabled the prediction of this effect in a molecule without nuclear calculations [3]. Recent measurements, combined with theoretical data, have made it possible to study the effect of the magnetization distribution of the Ra nucleus in the RaF molecule for the first time [4]. The use of the factorization method also makes it possible to directly extract nuclear magnetic moments and the parameters of their magnetization distributions from hyperfine-structure data for short-lived nuclei, for example Po [5]. The development of precision methods for calculating the atomic factors of field and mass shifts, including quantum electrodynamics effects, for the interpretation of isotope-shift measurements makes it possible to extract nuclear charge radii at a new level of accuracy with the uncertainty dominated by experiment [6]. This work was supported by Russian Science Foundation grant No. 26-12-00350 and BASIS Foundation grant No. 24-1-1-36-2. [1] L.V. Skripnikov, S.D. Prosnyak, Phys. Rev. C 2022, 106, 054303. [2] J.W. Blanchard, D. Budker, D. DeMille, M.G. Kozlov, L.V. Skripnikov, Phys. Rev. Research 2023, 5, 013191. [3] L.V. Skripnikov, J. Chem. Phys. 2020, 153, 114114. [4] S.G. Wilkins, S.M. Udrescu, M. Athanasakis-Kaklamanakis, R.F. Garcia Ruiz, M. Au, et al., Science 2025, 390, 386-389. [5] L.V. Skripnikov, A.E. Barzakh, Phys. Rev. C 2024, 109, 024315. [6] L.V. Skripnikov, S.D. Prosnyak, A. V. Malyshev, et al., Phys. Rev. A 2024, 110, 012807.