The Banach-Tarski paradox states that a ball can be disassembled into finitely many disjoint pieces and reassembled via translations and rotations into two copies of the original ball. It turns out that this "paradoxical decomposition" is precisely characterized by the group theoretic property known as (non)-amenability. Along the way, we will encounter various equivalent definitions of amenable group (of which there are many) and some applications. This talk will be accessible to anyone who knows what a group is.