ABSTRACT: Inpainting, the technique of modifying an image in an undetectable form, is as ancient as art itself. The goals and applications of inpainting are numerous, from the restoration of damaged paintings, photographs and films, to the removal of selected undesirable objects. This chapter is intended to present an overview of PDE based image inpainting algorithms, with emphasis in models developed by the authors. These models are based on the propagation of information along the image isophotes and on the minimization of an energy functional which follows a relaxation of the Elastica model. This last variational formulation can be easily extended to 3D to fill holes in surfaces, a problem closely related to image inpainting. Basic PDE-based approaches to inpainting share the shortcoming that they cannot restore texture, so combinations of these algorithms with texture synthesis techniques are also discussed. Some results are shown for applications such as removal of text, restoration of scratched photographs, removal of selected objects and reconstruction of 3D surfaces with holes. Other recent approaches to the image inpainting problem are also briefly reviewed.