Applications of top to random shuffles

The top to random shuffle is a method of shuffling cards which proceeds at each step by removing the top card and inserting it into a random position. We survey applications of the top to random shuffle to representation theory and combinatorics. Along the way we give an introduction to Stein's method (a remarkable technique for proving probabilistic limit theorems in situtations where exact enumeration fails). Finally, we give a new result on character ratios of symmetric groups. This talk should be accessible to a general audience.