MR 2827979 Reviewed Lando, S. K. Hurwitz numbers: on the edge between combinatorics and geometry. Proceedings of the International Congress of Mathematicians, volume IV, 2010, 2444--2470. (Reviewer Francesca Vetro) 14N35 (05A15 14H10 14H30 37K20)

Object of study in this paper are the Hurwitz numbers. They were introduced by Hurwitz in the end of nineteenth century and still they are of great interest. The Hurwitz numbers are important in topology because they enumerate ramified coverings of two-dimensional surfaces, but not only. The author observes that their importance in modern research is mainly due to their connections with the geometry of the moduli space of curves. Moreover, they are of interest in mathematical physics and group theory. The purpose of this paper is to describe the progress made in the last couple of decades in understanding Hurwitz numbers.