The differential Painlevé equations define the most general special functions (the Painlevé transcendents) and play an important role in modern mathematical physics. Regarding their applications in integrable systems, it turns out that the Painlevé equations can be obtained as reduced ODEs of some integrable PDEs.
In recent years, quantum or, more generally, non-abelian extensions of various integrable systems have acquired considerable attention. It was motivated by problematics and needs of modern quantum physics, as well as by natural attempts of mathematicians to extend and to generalize the ``classical'' integrable structures and systems. The Painlevé transcendents provide a good example of this phenomena, which we will look at in this talk.
