  This is the first of a series of papers on sheaf theory on smooth and
topological stacks and its applications. The main result of the present paper
is the characterization of the twisted (by a closed integral three-form) de
Rham complex on a manifold. As an object in the derived category it will be
related with the push-forward of the constant sheaf from a S^1-gerbe with
Dixmier-Douady class represented by the three-form. In order to formulate and
prove this result we develop in detail the foundations of sheaf theory for
smooth stacks.
