  Let V be a vertex operator algebra, and for k a positive integer, let g be a
k-cycle permutation of the vertex operator algebra V^{\otimes k}. We prove that
the categories of weak, weak admissible and ordinary g-twisted modules for the
tensor product vertex operator algebra V^{\otimes k} are isomorphic to the
categories of weak, weak admissible and ordinary V-modules, respectively. The
main result is an explicit construction of the weak g-twisted V^{\otimes
k}-modules from weak V-modules. For an arbitrary permutation automorphism g of
V^{\otimes k} the category of weak admissible g-twisted modules for V^{\otimes
k$ is semisimple and the simple objects are determined if V is rational. In
addition, we extend these results to the more general setting of \gamma
g-twisted V^{\otimes k}-modules for \gamma a general automorphism of V acting
diagonally on V^{\otimes k} and a g a permutation automorphism of V^{\otimes
k}.
