I will talk about a general formula relating the p-adic heights of Heegner points to derivatives of p-adic L-functions. It generalizes results of Perrin-Riou and Howard to the setting of the work of Yuan-Zhang-Zhang on the complex Gross-Zagier formula, and of Waldspurger's formula on toric periods. A special feature of the p-adic version is the possibility, first observed by Mazur, of an extension in which all objects vary analytically. I will explain the resulting picture and some applications.