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, 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.