Explore the representation theory of finite groups of Lie type in this 52-minute lecture from the Hausdorff Trimester Program on Logic and Algorithms in Group Theory. Delve into the structural properties of these groups and their relevance to representation theory. Examine fundamental goals and current research in the field. Learn about Harish-Chandra theory as a crucial tool, and focus on Deligne-Lusztig theory for classifying irreducible complex representations. Cover topics including finite classical groups, exceptional groups, finite reductive groups, the Lang-Steinberg theorem, maximal tori, Weyl groups, and Coxeter groups. Gain insights into the classification of finite simple groups and the orders of various finite groups of Lie type.