  Neutron stars (NSs) and black holes (BHs) are born when the final collapse of
the stellar core terminates the lives of stars more massive than about 9 Msun.
This can trigger the powerful ejection of a large fraction of the star's
material in a core-collapse supernova (CCSN), whose extreme luminosity is
energized by the decay of radioactive isotopes such as 56Ni and 56Co. When
evolving in close binary systems, the compact relics of such infernal
catastrophes spiral towards each other on orbits gradually decaying by
gravitational-wave emission. Ultimately, the violent collision of the two
components forms a more massive, rapidly spinning remnant, again accompanied by
the ejection of considerable amounts of matter. These merger events can be
observed by high-energy bursts of gamma rays with afterglows and
electromagnetic transients called kilonovae, which radiate the energy released
in radioactive decays of freshly assembled rapid neutron-capture elements. By
means of their mass ejection and the nuclear and neutrino reactions taking
place in the ejecta, both CCSNe and compact object mergers (COMs) are prominent
sites of heavy-element nucleosynthesis and play a central role in the cosmic
cycle of matter and the chemical enrichment history of galaxies. The nuclear
equation of state (EoS) of NS matter, from neutron-rich to proton-dominated
conditions and with temperatures ranging from about zero to ~100 MeV, is a
crucial ingredient in these astrophysical phenomena. It determines their
dynamical processes, their remnant properties even at the level of deciding
between NS or BH, and the properties of the associated emission of neutrinos,
whose interactions govern the thermodynamic conditions and the
neutron-to-proton ratio for nucleosynthesis reactions in the innermost ejecta.
This chapter discusses corresponding EoS dependent effects of relevance in
CCSNe as well as COMs. (slightly abridged)
