Painlevé Equations in the Differential Geometry of Surfaces 1. Introduction 2. Basics on Painlevé Equations and Quaternionic Description of Surfaces 3. Bonnet Surfaces in Euclidean Three space 4. Bonnet Surfaces in S3 and H3 and Surfaces with Harmonic Inverse Mean Curvature 5. Surfaces with Constant Curvature 6. Appendices.