In this paper we study some variational models for exemplar-based image inpainting, also referred to as nonlocal methods. Nonlocal methods for denoising and inpainting have gained considerable attention due to their good performance on textured images, a known weakness of classical local methods which are performant in recovering the geometric structure of the image. We rst review a general variational framework for the problem of nonlocal inpainting that exploits the self-similarity of natural images to copy information in a consistent way from the known parts of the image. We single out some particular methods and we review the main properties of the corresponding energies and their minima. We discuss the basic algorithms to minimize the energies and we display some numerical experiments illustrating the main properties of the proposed models.