Taussky-Todd Lecture

The celebrated Birch--Swinnerton-Dyer (BSD) conjecture connects the structure of the rational points on an elliptic curved defined over the rationals to the analytic properties of its associated Hasse--Weil L-function. In this talk I will recall the BSD conjecture (and its various parts) and related conjectures and survey some of the known results towards them, especially recent work. Much of this talk is intended for a general mathematical audience.