When a young mathematician told his advisor that he had found a way to turn a sphere inside out (without making any creases), his advisor told him that this was impossible to do so, and gave him a proof. His advisor was wrong. That young mathematician (Smale) went on to win the highest mathematical prize (the Fields Medal). His solution is very non-intuitive. Today, there are several excellent videos on YouTube that show how to turn a sphere inside out, and this talk will explain some of them.