  Neutrinos play a crucial role in the collapse and explosion of massive stars,
governing the infall dynamics of the stellar core, triggering and fueling the
explosion and driving the cooling and deleptonization of the newly formed
neutron star. Due to their role neutrinos carry information from the heart of
the explosion and, due to their weakly interacting nature, offer the only
direct probe of the dynamics and thermodynamics at the center of a supernova.
In this paper, we review the present status of modelling the neutrino physics
and signal formation in collapsing and exploding stars. We assess the
capability of current and planned large underground neutrino detectors to yield
faithful information of the time and flavor dependent neutrino signal from a
future Galactic supernova. We show how the observable neutrino burst would
provide a benchmark for fundamental supernova physics with unprecedented
richness of detail. Exploiting the treasure of the measured neutrino events
requires a careful discrimination of source-generated properties from signal
features that originate on the way to the detector. As for the latter, we
discuss self-induced flavor conversions associated with neutrino-neutrino
interactions that occur in the deepest stellar regions; matter effects that
modify the pattern of flavor conversions in the dynamical stellar envelope;
neutrino-oscillation signatures that result from structural features associated
with the shock-wave propagation as well as turbulent mass motions in post-shock
layers. Finally, we highlight our current understanding of the formation of the
diffuse supernova neutrino background and we analyse the perspectives for a
detection of this relic signal that integrates the contributions from all past
core-collapse supernovae in the Universe.
