  Neutrino oscillations in a hot and dense astrophysical environment such as a
core-collapse supernova pose a challenging, seven-dimensional flavor transport
problem. To make the problem even more difficult (and interesting), neutrinos
can experience collective oscillations through nonlinear refraction in the
dense neutrino medium in this environment. Significant progress has been made
in the last decade towards the understanding of collective neutrino
oscillations in various simplified neutrino gas models with imposed symmetries
and reduced dimensions. However, a series of recent studies seem to have
"reset" this progress by showing that these models may not be compatible with
collective neutrino oscillations because the latter can break the symmetries
spontaneously if they are not imposed. We review some of the key concepts of
collective neutrino oscillations by using a few simple toy models. We also
elucidate the breaking of spatial and directional symmetries in these models
because of collective oscillations.
