  We present a quantum field theoretical approach to the vacuum neutrino
oscillations in curved space, we analyze the non--trivial interplay between
quantum field mixing and field quantization in curved space and derive new
oscillation formulae. We compute the formulae explicitly in the spatially flat
FLRW metrics for universes dominated by a cosmological constant and by
radiation. We evaluate the transition probabilities in the Schwarzschild black
hole metric, and we show that the Hawking radiation affects the oscillations of
neutrinos. We show that our results are consistent with those of previous
analyses when the quantum mechanical limit is considered.
