ABSTRACT
Discrete Painlevé equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable n and there are three different types of equations according to whether the coefficient functions are linear, exponential or elliptic functions of n. In this chapter, we focus on the elliptic type and give a review of the construction of such equations on the E 8 lattice. The first such construction was given by Sakai [38]. We focus on recent developments giving rise to more examples of elliptic discrete Painlevé equations.
