Following Berger's holonomy classification and Atiyah and Donaldson's achievements in Yang-Mills theory, differential geometers have studied the interactions between variational and algebraic perspectives. Our project combines these traditions in the study of special geometric structures, such as extremal Kähler/Sasaki, special holonomy, generalized geometry and the interplay of all these concepts in Ströminger systems. In practice, these problems belong to gauge theory : a space of connections, a curvature equation to solve, a group of symmetries to control. In each case, the expected outcome is a correspondence between a special geometry and an algebraic condition, as provided by Kobayashi-Hitchin-D-U-Y, which allows to describe the local structure of the moduli space in terms of stability. In addition to constructing new families of examples, our objective is to understand their global topology, deformations and algebraic obstructions.