Explore advanced concepts in intertwining and strong stationary duality in this Institute for Mathematical Sciences Distinguished Visitor Lecture. Delve into Persi Diaconis' insights on constructing dual chains for random walks on groups with uniform stationary distributions. Examine applications in statistics that require less symmetrical constructions, focusing on Nestoridi's work on stationary times for random walks on hyperplane arrangement chambers and Igor Pak's examples of non-uniform duals. Gain a deeper understanding of current developments and unresolved questions in this complex mathematical field.