Solving equations in singular moduli amounts to finding so-called special points on algebraic varieties and is a simple instance of the André-Oort philosophy for Shimura varieties. Some of these equations are more interesting than others and have several consequences, which we shall review. Many results for singular moduli are expected to hold for differences of singular moduli as well, in the spirit of Gross and Zagier: we will focus in particular on linear equations and show that such differences are distinct. Joint work with Guy Fowler (Manchester).