Elliptic curves and explicit class field theory

The arithmetic of elliptic curves is related in multiple ways to explicit class field theory, notably through the theory of {\em complex multiplication}, one of the crown jewels of number theory. This connection plays a key role in recent progress in the recent theorem that there are positive proportions of elliptic curves of rank zero, and of rank one, for which the Birch and Swinnerton-Dyer conjecture is true, growing out of the work of Gross-Zagier, Kolyvagin, Bhargava-Shankar, Skinner-Urban-Wan, and Wei Zhang. I will discuss various relations that exist between elliptic curves and explicit class field theory, such as those above.