Georgia Institute of Technology College of Sciences

In late 1980's Manin et al put forward a precise conjecture about the density of rational points on Fano varieties. Over the last two decades some progress has been made towards proving this conjecture. But the conjecture is far from being proved even for the case of two dimensional Fano varieties or del Pezzo surfaces. These surfaces are geometrically classified according to `degree', and the geometric, as well as, the arithmetic complexity increases as the degree drops. The most interesting cases of Manin's conjecture for surfaces are degrees four and lower. In this talk I will mainly focus on the arithmetic of these del Pezzo surfaces, and report some of my own results (partly joint with Henryk Iwaniec). I will also talk about some other problems which apparently have a different flavor but, nonetheless, are directly related with the problem of rational points on surfaces.