We study asymptotics of the number of pairs of transverse simple closed curves living on a fixed smooth surface and having at most N intersections. The pairs of curves are considered up to a diffeomorphism of the surface. We perform separate count of pairs of positively intersecting simple closed curves and of pairs satisfying certain geometric restrictions. We also count pairs of transversally intersecting multicurves. We make special study of all these quantities in two opposite regimes: when the genus of the surface is very large and when the genus of the surface is equal to zero. In the latter case, pairs of transverse simple closed curves on a sphere correspond to classical meanders. The principal tools of our study are square-tiled surfaces and Masur-Veech volumes of the moduli spaces of Abelian and quadratic differentials. The talk is based on the ongoing project with V.Delecroix, E.Goujard and P.Zograf.