During their hydrostatic lives, stars generate energy by nuclear reactions. The most effective energies (the Gamow energy) at which these reactions occur in stars are so low that a direct measurement of the required cross sections is usually not possible, and data taken at higher energies have to be extrapolated to the Gamow energy. To reduce the uncertainty inherent in such extrapolations improved nuclear reaction models have to be developed, making use of the recent advances in nuclear structure theory. Some of the experimental low-energy cross sections are enhanced by screening effects due to the electrons present in target and/or projectile. A better understanding of such  effects has to be developed to remove the screening contaminations from the data.

For massive stars, the nuclear ashes produced during the hydrostatic burning stages are ejected out of the star's interior in supernova explosions. Despite impressive progress in supernova modeling, current computer simulations fail to explode. While an improved description of multidimensional effects (like convection, magnetic fields, rotation...) are needed in the models, there is also a quest for improved nuclear physics input. This includes improved descriptions of stellar weak-interaction rates, neutrino-nucleus reactions, the supernova equation of state (EoS) and neutrino opacities of dense and hot matter. These requirements can only be met by developing accurate nuclear structure theory and corresponding computational capability which can account for the relevant correlations among nucleons in the respective supernova environment.

For a wide range of core-collapse supernovae, a neutron star (or pulsar) is left behind after the explosion. Such neutron stars (can) serve as laboratory for nuclear physics at extreme conditions in density and isospin. At the birth of the neutron star the matter is also quite hot and for a reliable description of the cooling phase calculations of the EoS and the neutrino opacities  have to be improved by using more realistic strong interactions which, in particular, include the tensor correlations among nucleons and by accounting for the in-medium  renormalization of the weak interaction. Improved theory that can accurately calculate and predict nuclear masses, in particular, at large neutron to proton ratios, are needed for a reliable description of the outer crust of a neutron star and its matter composition, which is essential for the correct interpretation of surface temperature data of cooling neutron stars, obtained from X-ray observations. More reliable nuclear interactions and more precise many-body methods are required to describe the inner crust of the neutron star. The core EoS is essential for the determination of the maximum mass of neutron stars. Current theories have to be improved by considering 3-body and, possibly, 4-body forces and relativistic effects as well as the role of hyperons in the core EoS.

The elements in the Universe with A > 60 are made by the neutron capture reactions, one half in the s-process during stellar He burning on a time scale slow compared to beta-decay and the other half by the r-process, a sequence of rapid neutron captures interrupted by beta-decays. For the s-process, neutron capture cross sections constitute the essential nuclear input, including capture on radioactive nuclei, and give rise to branchings in the reaction path. Although the actual astrophysical r-process site is not yet known, there is general agreement that the r-process has to occur under explosive conditions (supernovae, neutron star mergers) where a large amount of neutrons is available for a short time. As the r-process path runs through extremely neutron rich nuclei, far away from the valley of stability, most properties of the nuclei on the r-process path are not known experimentally and have to be calculated. The most important nuclear ingredients in r-process simulations are neutron separation energies (masses), half-lives, fission probabilities for the heaviest nuclei to include possible recycling and, after freeze-out, neutron capture cross sections. Similarly, description of the p-process, which accounts for the rare proton rich isotopes, requires a comprehensive set of (gamma,n), (gamma,alpha), and (gamma,p) cross sections and of the respective data for the inverse reactions. To derive such ingredients for these nucleosynthesis processes global nuclear models with reliability at extreme neutron-to-proton ratios are needed. As these ingredients have been so far derived from empirical models, the next step is now to determine them from microscopic models which are globally applicable, are based on well-controlled and understood nucleon-nucleon interactions and take relevant correlations among nucleons into account.

Explosive hydrogen burning occurs in certain binary systems when there is mass accretion from a companion star on a compact object. The relevant nuclear processes are proton- and alpha-induced reactions running through proton-rich nuclei far from stability. Often, the relevant rates are determined by a few individual nuclear states. This prohibits the application of statistical models. Only in a few instances direct measurements of reaction cross sections are possible. Therefore nuclear reliable theory is needed to extract information from indirect measurements (Coulomb dissociation, transfer reactions etc.) or to supply the required information (e.g. energy levels and spins, spectroscopic factors etc.) whenever data are not available. In x-ray bursters the reaction flow to higher masses is hindered by certain waiting point nuclei like Ge-64, Se-68, and Kr-72 where beta half-lives are long and proton captures lead to unbound compound nuclei. Here modern 3-body models, as developed for halo nuclei, have to be employed to determine whether these waiting points can be bridged by sequential 2-proton captures. Modern nuclear structure theory is needed to determine the changes of nuclear properties due to the stellar environment, e.g. stellar half-lives due to thermal population of excited nuclear levels.