Title: Dimension distortion of images of sets under Sobolev mappings. Abstract: Suppose that a set $M\subset \mathbb R^n$ has Hausdorff dimension $\alpha$ and a continuous mapping $f$ is in Sobolev space $W^{s,p}$. We give a sharp estimate for the dimension of the image $f(M)$ in terms of $\alpha, n, s$ and $p.$ We also study the subcritical mappings.