Chris Skinner of Princeton University revisits one of the Clay Millennium Prize problems in this Institute for Advanced Study members' colloquium. He restates the Birch and Swinnerton-Dyer Conjecture, which links the rank of an elliptic curve's rational points to the behavior of its associated L-function at a special point, and surveys the evidence built up for it over decades. The talk traces progress from the conjecture's origins through classical results tying rank to analytic behavior, into more recent advances made after 2000, largely repeating the plenary talk Skinner gave at the 2025 Clay Research conference on the Millennium Prize problems. It is aimed at a mathematically literate audience familiar with number theory, walking through why the conjecture matters, what has been proven in special cases, and where the general problem still stands open.