*Ranks of elliptic curves*

**Abstract:** Elliptic curves have surprising and important applications both
inside and outside of number theory. Elliptic curves over
finite fields have cryptographic applications, while elliptic
curves over the rational numbers were used to prove Fermat's
Last Theorem. The rank of an elliptic curve is a measure of
the number of rational points on the curve. We will explore
questions about the distribution of ranks, especially in families
of quadratic twists.