  Neutrino masses and mixings produce vacuum oscillations, an established
quantum mechanical phenomenon. In matter, the Mikheev-Smirnov-Wolfenstein
effect, due to neutrino interactions with the background particles, triggers
resonant flavor modification. In dense environments, such as core-collapse
supernovae or compact mergers, sizable neutrino-neutrino interactions, shock
waves and turbulence impact the neutrino flavor content under a variety of
phenomena. Theoretical approaches of neutrino propagation range from the
mean-field approximation to the full quantum kinetic equations. Intriguing
connections have been uncovered between weakly interacting dense neutrino gases
and other many-body systems and domains, from condensed matter and nuclear
physics to quantum computing. Besides the intrinsic theoretical interest,
establishing how neutrinos change flavor contributes to answer the longstanding
open questions of how massive stars explode and of the r-process sites. It is
also important for future observations of core-collapse supernova neutrinos and
of the diffuse supernova neutrino background that should be discovered in the
foreseeable future.
