It is often the case that in concrete Diophantine problems it is not enough to show that the rational points on a variety are algebraically degenerate; one also needs to describe their Zariski closure. In arithmetic questions such as Vojta's conjecture and the Bombieri-Lang conjecture, this Zariski closure is referred to as the special set. In this talk I will explain some techniques based in omega-integrality to describe the special set in Diophantine conjectures over number fields and function fields, and to prove some instances of these conjectures in the function field case.