In this talk we look at algebraic relations, over C(t), between solutions (and derivatives) of the Painlevé equations. I will explain how one can prove that the algebraic independence conjecture holds, namely: if y_1,…,y_n are distinct solutions of a generic Painlevé Equations, then y_1,y'_1,…,y_n,y'_n are algebraically independent over  C(t).

I will explain the role of differential Galois theory as well as discuss how one can reduce the problem to studying the Riccati equations.