As rigorous methodology for studying the Riemann-Hilbert problems associated with certain integrable nonlinear ODEs was introduced in 1992 by Fokas and Zhou, and was used to investigate Painleve II and Painleve IV equations. Here the authors apply this methodology to Painleve I, III, and V equations. They show that the Cauchy problems for these equations admit in general global solutions, meromorphic in t. Furthermore, for special relations among the monodromy data and for t on Stokes lines, these solutions are bounded for finite t. In connection with Painleve I they note that the usual Lax pair gives rise to monodromy data some of which depend nonlinearly on the unknown solution of Painleve I. This problem is bypassed here by introducing a new Lax pair for which all the monodromy data are constant.