A square-tiled surface is a ramified cover of a torus branched over a single point. The mapping class group of the torus is $SL(2,Z)$ and it acts on such covers by the change of the base torus. We would like to understand orbits of this action. Even in the simplest case when the genus of the square-tiled surfaces is 2 the answer is not known. I will speak on advances on the question in this case.