Abstract: Growing up, I always loved learning about world-changing scientific breakthroughs that were discovered by accident. Penicillin, artificial sweeteners, X-rays, and synthetic dyes are just a few of the discoveries that were stumbled upon by scientists who had other goals in mind. More recently, I have come to wonder why anecdotes about accidental discoveries in mathematics are not as commonplace. Is it a fundamental difference in they way mathematicians and natural scientists view their work? Are such stories too contrary to the popular perception that success in mathematics is reserved for the genius of a select few? Whatever the reason, I argue that mathematics happens accidentally all the time. In this talk, I will describe two accidental discoveries from my own work involving Penrose tilings, circle packings, chordal graphs, lecture hall partitions, lattice polytopes, and polynomial rings.