It is a classical problem to try to count the number of closed curves on (hyperbolic) surfaces with bounded length. Due to people such as Delsart, Huber, and Margulis it is known that the asymptotic growth of the number of curves is exponential in the length. On the other hand, if one only looks at simple curves the growth is polynomial. Mirzakhani proved that the number of simple curves on a hyperbolic surface of genus g of length at most L is asymptotic to L^{6g-6}. Recently, she extended her result to also hold for curves with bounded self intersection, showing that the same polynomial growth holds. In this talk I will discuss her results and some recent generalizations that shows that the same asymptotic also holds for other metrics on the surface.

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