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The goal of this work is to construct, for a smooth variety $X$ over a perfect field k of finite characteristic $p\>0$, an overconvergent de Rham-Witt complex $W^{†}{\Omega }_{X/k}$ as a suitable subcomplex of the de Rham-Witt complex of Deligne-Illusie. This complex, which is functorial in $X$, is a complex of étale sheaves and a differential graded algebra over the ring $W^{†}\left({𝒪}_X\right)$ of overconvergent Witt-vectors. If $X$ is affine one proves that there is an isomorphism between Monsky-Washnitzer cohomology and (rational) overconvergent...