Title: Counting rational points on cubics and quartics.

Abstract: While the problem of counting the number of rational points on del Pezzo surfaces of high degrees has been successfully tackled using geometric techniques, that of lower degrees still remains out of reach of the present methods. In this talk, I will describe a joint work with Henryk Iwaniec, where we use a combination of sieve methods and techniques from analytic number theory to get bounds for the number of rational points on del Pezzo surfaces of degree 3 (i.e. cubic surfaces in