With the much-anticipated multi-petawatt (PW) laser facilities that are coming online, neutron sources with extreme fluxes could soon be in reach. Such sources would rely on spallation by protons accelerated by the high-intensity lasers. These high neutron fluxes would make possible not only direct measurements of neutron capture and β-decay rates related to the r-process of nucleosynthesis of heavy elements, but also such nuclear measurements in a hot plasma environment, which would be beneficial for s-process investigations in astrophysically relevant conditions. This could, in turn, finally allow possible reconciliation of the observed element abundances in stars and those derived from simulations, which at present show large discrepancies. Here, we review a possible pathway to reach unprecedented neutron fluxes using multi-PW lasers, as well as strategies to perform measurements to investigate the r- and s-processes of nucleosynthesis of heavy elements in cold matter, as well as in a hot plasma environment.