  Let $L$ be a line bundle on a K3 or Enriques surface. We give a vanishing
theorem for $H^1(L)$ that, unlike most vanishing theorems, gives necessary and
sufficient geometrical conditions for the vanishing. This result is essential
in our study of Brill-Noether theory of curves on Enriques surfaces (reference
[KL1]) and of Enriques-Fano threefolds (reference [KLM]).
