A singular modulus is the j-invariant of an elliptic curve with complex multiplication. Given a singular modulus x we denote by \Delta_x the discriminant of the associated imaginary quadratic order. We denote by h(\Delta) the class number of the imaginary quadratic order of discriminant \Delta. Recall that two singular moduli x and y are conjugate over Q if and only if \Delta_x = \Delta_y , and that all singular moduli of a given discriminant \Delta form a full Galois orbit over Q. In particular, [Q(x) : Q] = h(\Delta x). Here, we show that the field Q(x, y), generated by two singular moduli x and y, is generated by their sum x + y, unless x and y are conjugate over Q, in which case x + y generates a subfield of degree at most 2. We obtain a similar result for the product of two singular moduli. Futhermore, we fix a rational number \alpha \neq 0, \pm 1 and show that the field Q(x, y) is generated by x + \alpha y, with a few exceptions occurring when x and y generate the same quadratic field over Q. These are the results from my collaborations with Antonin Riffault, Yuri Bilu and Huilin Zhu.