After a successful core collapse supernova (CCSN) explosion, a hot dense proto-neutron star (PNS) is left as a remnant. Over a time of 20 or so seconds, this PNS emits the majority of the neutrinos that come from the CCSN, contracts, and loses most of its lepton number. This is the process by which all neutron stars in our galaxy are likely born. The emitted neutrinos were detected from supernova (SN) 1987A, and they will be detected in much greater numbers from any future galactic CCSN. These detections can provide a direct window into the properties of the dense matter encountered inside neutron stars, and they can affect nucleosynthesis in the material ejected during the CCSN. In this chapter, we review the basic physics of PNS cooling, including the basic equations of PNS structure and neutrino diffusion in dense matter. We then discuss how the nuclear equation of state, neutrino opacities in dense matter, and convection can shape the temporal behavior of the neutrino signal. We also discuss what was learned from the late-time SN 1987A neutrinos, the prospects for detection of these neutrinos from future galactic CCSNe, and the effects these neutrinos can have on nucleosynthesis
