Supernova neutrino oscillations: what do we understand?

We summarize our current understanding of the neutrino flavor conversions inside a core collapse supernova, clarifying the important role played by the "collective effects" in determining flavor conversion probabilities. The potentially observable $\nu_e$ and $\bar{\nu}_e$ spectra may help us identify the neutrino mixing scenario, distinguish between primary flux models, and learn more about the supernova explosion.