  Let $V$ be a vertex operator algebra and $g$ an automorphism of finite order.
We construct an associative algebra $A_g(V)$ and a pair of functors between the
category of $A_g(V)$-modules and a certain category of admissible $g$-twisted
$V$-modules. In particular, these functors exhibit a bijection between the
simple modules in each category. We give various applications, including the
fact that the complete reducibility of admissible $g$-twisted modules implies
both the finite-dimensionality of homogeneous spaces and the finiteness of the
number of simple $g$-twisted modules.
