In this series of two lectures I will review what we know about moduli space of instanton bundles on projective spaces. We will see that the situation drastically change when the dimension of the projective space increase. At the end of the two talks I will state some interesting open problems.

This course is organized within the framework of Project "Geometria Algebrica em Portugal"(PTDC/MAT/099275/2008) and Project "Topological Invariants via Differential Geometry" (PTDC/MAT/098770/2008).