On the conjecture of Birch and Swinnerton-Dyer

The conjecture of Birch and Swinnerton-Dyer, which relates the rank of an elliptic curve over the rational numbers to the order of vanishing of its L-function at the point s = 1, has been a central problem in number theory for over fifty years. In this talk, which is aimed at a general mathematical audience, I will review the theorem of Mordell and Weil on the group of rational points and give the definition of the L-function, before presenting the general conjecture. I will then describe some of the progress that has been made towards its resolution.