The semigeostrophic equations are a set of equations which model large-scale atmospheric/ocean flows. The system admits a dual version, obtained from the original equations through a change of variable. Existence for the dual problem has been proven in 1998 by Benamou and Brenier, but the existence of a solution of the original system remained open due to the low regularity of the change of variable. In the talk we prove the existence of distributional solutions of the original equations, both in R^3 and in a two-dimensional periodic setting. The proof is based on recent regularity and stability estimates for Alexandrov solutions of the Monge-Amp`ere equation, established by De Philippis and Figalli.