**4. Summary**

Due to the extreme densities, temperatures or neutron excesses encountered in astrophysical environments, the properties of nuclei cannot be measured directly in a laboratory and have to be modeled. If these properties are strongly influenced by nucleon correlations, the diagonalization shell model is the method of choice. In recent years, such studies have been performed to derive the electron capture rates and neutrino-induced cross sections for nuclei in the *sd*- and *p f*-shell advancing our understanding of the core evolution of intermediate-mass and massive stars. Another important application of the diagonalization shell model was the calculation of half-lives for rapid neutron-capture process (r-process) nuclei with magic neutron numbers, which serve as waiting points for the r-process' mass flow. This example also shows the limitation of current shell model applications as such studies would be also very desirable for the other nuclei on the r-process path, but cannot be performed yet as the required model spaces exceed current computational possibilities. These limitations in model space can be overcome within the Shell Model Monte Carlo (SMMC) approach, which is an alternative formulation of the shell model. This approach describes nuclear properties at finite temperature, but is not capable of detailed spectroscopy. Thus, the SMMC cannot be used to calculate r-process half-lives, which need a state-by-state description of transition strength. However, the ability of the SMMC approach to describe nuclear properties at finite temperatures including correlations paves the way to determine electron capture rates of heavier nuclei, which are crucial for the fate of core-collapse supernovae. In particular, SMMC allows the evaluation of how the Pauli blocking of Gamow–Teller strength at closed shells is overcome by correlations. On the basis of these studies, it could be demonstrated that neither *N* = 40 nor *N* = 50 neutron shell closure serve as severe obstacles for electron capture on nuclei. It is now commonly accepted that electron capture proceeds on nuclei throughout the entire collapse.

**Author Contributions:** writing—original draft preparation, K.L. and G.M.-P.; writing—review and editing, K.L. and G.M.-P.; All authors have read and agreed to the published version of the manuscript. **Funding:** This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project-ID 279384907, SFB 1245 *Nuclei: From Fundamental Interactions to Structure and Stars*, the Helmholtz Forschungsakademie Hessen für FAIR and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERC Advanced Grant KILONOVA No. 885281).

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors are grateful for the long-term collaboration with the members of the Strasbourg-Madrid shell model group. We have learnt a lot from Etienne Caurier, Frederic Nowacki, Alfredo Poves and Andres Zuker. Our work has also benefited strongly from collaborations with many astrophysicists, most notably with Raphael Hix, Hans-Thomas Janka and Friedel Thielemann.

**Conflicts of Interest:** The authors declare no conflict of interest.
