This paper is an expository account of the Banach-Tarski Paradox, written for my Minor Thesis requirement of the Ph.D. program at Harvard University. The purpose of the Minor Thesis was to pick a subject that you know little about, and spend a 3-week period learning and writing an account of the subject. I wrote this to be readable by an advanced undergraduate or first year graduate student in mathematics. I'm interested in comments; tell me if you like it!