This is the first of two papers on the de Rham-Witt complex of a smooth variety over a perfect field of positive characteristic, introduced by Bloch-Deligne-Illusie. The present part deals with the problem of duality. It is shown that the top degree part of the dRW complex truncated at level n is a dualizing module and dualizing complex for the Witt vector scheme of level n of the variety. Furthermore, the de Rham- Witt complex of level n turns out to be self dual. The auto duality for the cohomology of the level n dRW of a smooth and proper variety follows directly. When one passes to the limit over n an autoduality for the dRW complex as a complex over the Raynaud ring is obtained. A major part of the paper is concerned with the algebraic properties of the dualizing functor on complexes of coherent modules over the Raynaud ring, the goal has been to obtain results with which one can compute. The paper finishes with two examples.