Gluing the opposite sides of the square one obtains a torus endowed with a flat metric (inherited from the Euclidean metric of the plane). Similarly one can build surfaces of larger genus by gluing parallel sides of several squares. These "square-tiled surfaces" are naturally endowed with a flat metric with conical singularities. In this talk I will present some recent results and conjectures about the geometry of these surfaces (and bigger families of flat surfaces) in large genus (joint work with V. Delecroix, P.Zograf and A. Zorich). These results can also be interpreted in terms of multicurves on surfaces and meanders.