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The classical Painlev\'e equations are so well known that it may come as asurprise to learn that the asymptotic description of its solutions remainsincomplete. The problem lies mainly with the description of families ofsolutions in the complex domain. Where asymptotic descriptions are known, theyare stated in the literature as valid for large connected domains, whichinclude movable poles of families of solutions. However, asymptotic analysisnecessarily assumes that the solutions are bounded and so these domains must bepunctured at locations corresponding to movable poles, leading to asymptoticresults that may not be uniformly valid. To overcome these issues, we recentlycarried out asymptotic analysis in Okamoto's geometric space of initial valuesfor the first and second Painlev\'e equations. In this paper, we review thismethod and indicate how it may be extended to the discrete Painlev\'eequations.

An Overview of Geometric Asymptotic Analysis of Continuous and Discrete Painlev\'e Equations