The authors analyze the global asymptotic properties of Painlevé equations, by means of the isomonodromy method. Their goal is to provide a rigorous justification for the asymptotic analysis of these equations. There are three approaches to be considered; (i) local asymptotic analysis of the Painlevé equation, (ii) the method of Kitaev based on the Brouwer fixed point theorem, and (iii) the method of Deift and Zhou, based on direct asymptotic analysis of the relevant oscillatory Riemann- Hilbert problems. Examples include the connection problem for pure imaginary solutions of the second Painlevé equation
. There is an interesting and useful analysis of the relation between the Deift-Zhou steepest descent method, and the Kitaev method.