Number Theory Seminar: The quest for elliptic curves of high rank

In 1922 Mordell proved that $E(\mathbb{Q})$ is a finitely-generated abelian group for every elliptic curve $E$ over $\mathbb{Q}$.  Mazur, 55 years later, determined the list of all torsion groups $T$ that can appear.  We still do not know which ranks $r$ are possible, or even whether we should expect $r$ to be bounded.

One way to try to get a sense of the answer is trying hard to find examples of curves of large rank.  We describe the history, diverse ingredients, and current state of the ongoing quest to find curves $E$ that have large $r$, or whose rank is large given a condition such as the structure of $T$ or the size (conductor, height, etc.) of $E$.