Explore the classification of simple amenable C*-algebras in this 47-minute lecture by Stuart White for the International Mathematical Union. Delve into operator algebras, examining structure and classification through examples from group actions and inductive limits. Investigate the role of traces as non-commutative invariant measures and the Elliott programme. Analyze higher-dimensional examples, considering topological dimensions and non-commutative tensor units. Gain insights into the fundamental aspects of C*-algebra structure and classification, building upon concepts from von Neumann algebras and Connes' work.