Studying image inpainting and denoising based on some partial differential equations (PDEs) is the main goal of this thesis. Among several approaches have been proposed to solve the problem, PDE-based methods are the most powerful ones. Image inpainting means to restore a damaged or corrupted image that information of some parts of it has been lost or changed. Inpainting may also be a useful tool for some people who artificially need either to remove some parts of an image such as overlapping texts, or to implement tricks used in special effects. It is important to restore the missing parts of an image so that the final image looks unaltered to the naked eye. Denoising is to remove noisy components from the pixels of an image. We must note that in common image enhancement applications, the pixels contain both information about the real data and the noise (e.g., image plus noise for additive noise), while in image inpainting there is no significant information in the region to be inpainted. Recently, a new PDE approach to image inpainting and image denoising are presented by Barbu et al. To the best of our knowledge, the pioneer work of Bertalmio et al. in 2000 is the first approach of image inpainting that introduced a third-order PDE that propagates the level lines arriving at hole. And numerous nonlinear PDE denoising approaches based on diffusion have been introduced since the early work of Perona and Malik in 1987. In this thesis at first, some preliminaries which will be used in the sequel are prepared. Then, we investigate a PDE-based approach to image inpainting using a minimization problem. It must be interesting that the history of image inpainting returns to about recent 20 years. We provide important theories about the resulting PDE, and solve it using a ltr"