This talk will form part of a series of three talks focusing on Broué’s Abelian Defect Group Conjecture, which concerns the modular representation theory of finite groups. We will pay particular attention here to the ‘geometric’ form of the conjecture which concerns finite reductive groups such as $GL_n(q)$ and $SL_n(q)$. Broué’s conjecture gives a strong structural reason for many numerical coincidences one sees amongst characters and is part of a general ‘local/global phenomena’ that is abundant in the theory. In this first talk we will briefly recall the necessary background material, get to the point where we can state the conjecture, and discuss some important examples.