  The classical Painlev\'e equations are so well known that it may come as a
surprise to learn that the asymptotic description of its solutions remains
incomplete. The problem lies mainly with the description of families of
solutions in the complex domain. Where asymptotic descriptions are known, they
are stated in the literature as valid for large connected domains, which
include movable poles of families of solutions. However, asymptotic analysis
necessarily assumes that the solutions are bounded and so these domains must be
punctured at locations corresponding to movable poles, leading to asymptotic
results that may not be uniformly valid. To overcome these issues, we recently
carried out asymptotic analysis in Okamoto's geometric space of initial values
for the first and second Painlev\'e equations. In this paper, we review this
method and indicate how it may be extended to the discrete Painlev\'e
equations.
