Almost since their very discovery over a century ago, it is known that the classical Painlevé equations govern monodromy preserving deformations of certain linear ODEs. This lies at the heart of the powerful Riemann-Hilbert approach to these equations. In this talk, I will discuss recent extensions of this approach to the q-difference setting, focusing on the q-analog of Painlevé VI derived by Jimbo and Sakai. I will show how, analogous to the classical theory, a corresponding monodromy manifold can be constructed and the global asymptotics of solutions can be derived by analysing associated Riemann-Hilbert problems. The special role of classical-function solutions in this framework will also be highlighted.

This is based on joint work with Nalini Joshi.