Abstract: I will start by reviewing some recent results on qualitative properties of local minimizers of the interaction energy to motivate the main topic of my talk: to discuss global minimizers. We will show the existence of compactly supported global miminizers under quite mild assumptions on the potential in the complementary set of classical H-stability in statistical mechanics. A strong connection with the classical obstacle problem appears very useful when the singularity is strong enough at zero. We will also discuss some aspects of the problem with nonlinear diffusion terms in the diffusive dominated regime where we show the existence of equilibria.