Abstract: The goal of this talk is to describe some interesting geometry that underlines a special class of discrete integrable systems known as discrete Painlevé equations. In the first hour we'll consider some concrete examples that we then use as a motivation for the general introduction into the Sakai's theory of classification of discrete Painlevé equations using the Okamoto surface of the equation. In the second hour we'll describe various sources of such equations and then show how the underlying geometry can help us relate different equations of the same type, up to finding very explicit change of coordinates that transforms one equation into the other.
