Square-tiled surfaces, Masur-Veech volumes and count of meanders

(joint work with V. Delecroix, E. Goujard and P. Zograf)
I will introduce square-tiled surfaces and explain why they represent integer points in the moduli spaces of Abelian and quadratic differentials. I will also explain why count of square-tiled surfaces provides Masur-Veech volumes of these moduli spaces.
To justify my interest to count of square-tiled surfaces I will show how it allows to count meanders. The results are obtained jointly with V. Delecroix, E. Goujard, P. Zograf.