The purpose of this monograph is two-fold: it introduces a conceptual language for the
geometrical objects underlying Painlevé equations and it offers new results on a particular
Painlevé III equation of type PIII (D6) called PIII (0 0 4 -4) describing its relation to
isomonodromic families of vector bundles on P1 with meromorphic connections. This equation is
equivalent to the radial sine (or sinh) Gordon equation and as such it appears widely in
geometry and physics. It is used here as a very concrete and classical illustration of the
modern theory of vector bundles with meromorphic connections. Complex multi-valued solutions on
C* are the natural context for most of the monograph but in the last four chapters real
solutions on R>0 (with or without singularities) are addressed. These provide examples of
variations of TERP structures which are related to tt geometry and harmonic bundles. As an
application a new global picture o0 is given.
