  We develop a theory of affine flag varieties and of their Schubert varieties
for reductive groups over a Laurent power series local field k((t)) with k a
perfect field. This can be viewed as a generalization of the theory of affine
flag varieties for loop groups to a "twisted case"; a consequence of our
results is that our construction also includes the flag varieties for Kac-Moody
Lie algebras of affine type. We also give a coherence conjecture on the
dimensions of the spaces of global sections of the natural ample line bundles
on the partial flag varieties attached to a fixed group over k((t)) and some
applications to local models of Shimura varieties.
