Heun's differential equation and its q-deformation

The $q$-Heun equation is a $q$-difference analogue of Heun’s differential equation. We review several solutions of Heun’s differential equation and investigate polynomial-type solutions of $q$-Heun equation. The limit $q\to 1$ corresponding to Heun’s differential equation and the ultradiscrete limit $q\to 0$ are considered.