What are all rational number solutions to the equation y² = x³ + 17? An easy rational solution is (x,y) = (2,5). Other rational solutions are less apparent, such as (-64/25,59/125) and (5023/3249,-842480/185193)! How were they found?

The graph of an equation like y² = x³ + 17 is called an elliptic curve, and such curves are a central topic in number theory. In recent years elliptic curves played a crucial role in the proof of Fermat's Last Theorem, and they are of great use in cryptography. We'll give an overview of elliptic curves, discussing some of their main properties and ending with several open questions.