What are the birational involutions of the plane, up to conjugation? They were classified by A. Beauville and L. Bayle in 2000 over algebraically closed fields. In a collaboration with I. Cheltsov, F. Mangolte and E. Yasinsky, we classified the ones of the real plane. In this talk I will present part of the classification (it is too long) and talk about some of the interesting phenomenon that may happen over the real numbers that do not happen over the complex numbers.