Nuclear astrophysics is that branch of astrophysics which helps understanding the Universe, or at least some of its many faces, through the knowledge of the microcosm of the atomic nucleus. It attempts to find as many nuclear physics imprints as possible in the macrocosm, and to decipher what those messages are telling us about the varied constituent objects in the Universe at present and in the past. In the last decades much advance has been made in nuclear astrophysics thanks to the sometimes spectacular progress made in the modelling of the structure and evolution of the stars, in the quality and diversity of the astronomical observations, as well as in the experimental and theoretical understanding of the atomic nucleus and of its spontaneous or induced transformations. Developments in other sub-fields of physics and chemistry have also contributed to that advance. Notwithstanding the accomplishment, many longstanding problems remain to be solved, and the theoretical understanding of a large variety of observational facts needs to be put on safer grounds. In addition, new questions are continuously emerging, and new facts endangering old ideas.This review shows that astrophysics has been, and still is, highly demanding to nuclear physics in both its experimental and theoretical components. On top of the fact that large varieties of nuclei have to be dealt with, these nuclei are immersed in highly unusual environments which may have a significant impact on their static properties, the diversity of their transmutation modes, and on the probabilities of these modes. In order to have a chance of solving some of the problems nuclear astrophysics is facing, the astrophysicists and nuclear physicists are obviously bound to put their competence in common, and have sometimes to benefit from the help of other fields of physics, like particle physics, plasma physics or solid-state physics. Given the highly varied and complex aspects, we pick here some specific nuclear physics topics which largely pervade nuclear astrophysics. Direct cross-section measurements 7.3.2. Indirect cross-section measurements 7.4. Neutron capture reactions: Experiments 7.5. Thermonuclear reaction rates: Models 7.5.1. Microscopic models 7.5.2. The potential and DWBA models 7.5.3. Parameter fits 7.5.4. The statistical models 8. Selected topics 8.1. Heavy-element nucleosynthesis by the s-and r-processes of neutron captures 8.1.1. Defining the s-process 8.1.2. Defining the r-process 8.1.3. The s-and r-process contributions to the solar-system composition 8.1.4. Astrophysical sites for the s-and r-processes 8.1.5. Heavy elements in low-metallicity stars 8.2. Cosmochronometry 8.2.1. Nucleo-cosmochronology: generalities 8.2.2. The trans-actinide clocks 8.2.3. The 187 Re -187 Os chronometry 8.3. Type-II supernovae 8.3.1. Evolution of massive stars leading to neutrino-driven supernovae 8.3.2. Nucleosynthesis in the hot bubble: Can the r-process occur ? 8.3.3. Signatures of a large-scale mixing of nucleosynthesis products 9. Summary References * The situ...