In diophantine geometry, the Batyrev-Manin-Peyre conjecture originally concerns the distribution of rational points on Fano varieties. It describes the asymptotic behaviour of the number of rational points of bounded height, when the bound becomes arbitrary large. 

A geometric analogue of this conjecture deals with the asymptotic behaviour of the moduli space of rational curves on a complex Fano variety, when the « degree » of the curves « goes to infinity ». Various examples support the claim that, after renormalisation in a relevant ring of motivic integration, the class of this moduli space may converge to a constant which has an interpretation as a motivic Euler product. 

In this talk, we will state a motivic version of the Batyrev-Manin-Peyre conjecture and give some examples for which it is known to hold : projective space, more generally toric varieties, and equivariant compactifications of vector spaces. In time permits, we will then discuss the more general notion notion of equidistribution of curves.