  Assuming two deep but standard conjectures from the Langlands Programme, we
prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic
fields Q(\sqrt{-d}) with -d=2, 3 mod 4. For a general number field K, again
assuming standard conjectures, we give a criterion based on the solutions to a
certain S-unit equation, which if satisfied implies the asymptotic Fermat's
Last Theorem.
