In the late 1980s Manin and others initiated a program to count rational points of bounded height on varieties defined over number fields. Looking at those varieties whose rational points are Zariski dense, conjectural asymptotic formula were predicted. An interesting test class of varieties consists of the (generalized) del Pezzo surfaces. We will outline the program in progess to prove Manin's conjecture for this class of varieties. This involves descent to integral points on a certain quasiaffine variety and techniques to approximate sums over these integral points by certain natural adelic integrals. Throughout the study of this arithmetic problem, geometric ideas are the constant source of inspiration. The speaker aspires to make the talk accessible to a wide audience, including especially graduate students.