The theory of Heegner points, which form a supply of algebraic points on a given elliptic curve, are one of the main tools used in proving the known cases of the Birch and Swinnerton-Dyer conjecture.
These points, which are attached to (orders of) imaginary quadratic fields K, arise from CM points on the modular curve. At the turn of the century H.Darmon introduced a construction of points attached to real quadratic fields, but these points were defined over p-adic fields and were only conjectured to be algebraic. In this talk I will give an overview of the generalizations and results that have been developed in the past two decades, and the contributions that we have made in the more recent years. This is joint work with Xavier Guitart.