Elliptic curves are a class of cubic curves (defined by a cubic polynomial in two variables) with deeply surprising and beautiful properties. The most surprising fact about elliptic curves is that their points naturally form an additive structure under some geometric operation. Elliptic curves show up everywhere in mathematics from tori defined over the complex numbers to generating abelian extensions of number fields. Elliptic curves also feature prominently in the Birch and Swinnerton-Dyer conjecture, one of the Millenium prize problems, and Andrew Wiles' proof of Fermat's last theorem. In this class, we will develop some fundamental results about elliptic curves such as Weierstrass function theory, the group law, and modular forms before diving head-first into some more advanced territory mentioned above. This class will be a meandering relentless rollercoaster through some of the most beautiful connections in modern mathematics rather than a reasonably-paced, well-structured, and rigorous development of a topic. Expect proofs to be "sketched", definitions to be hand waved, and lots of inane terminology to be used. That said, expect to come away with a broader awareness of open problems in mathematics, a much richer appreciation of the interplay between geometry and algebra, and a healthy respect for cubic polynomials in two variables.