On the bounded negativity conjecture

Abstract. In my talk I would like to present an outline of recent developments on the bounded negativity conjecture. In the first part I will introduce the notion of Harbourne constants and I will present  effective bounds for these constant for some classes of algebraic surfaces, mostly for blow ups of the complex projective plane. Secondly, I will show that, somehow surprisingly, the bounded negativity can be studied from the perspective of Zariski decompositions. If time will permit, I would like also to say a few words about the geography problem of surfaces of general type related to the bounded negativity problem.