This chapter reviews some interesting partial differential equations (PDEs) to motivate our readers. It provides a very quick review of several popular numerical methods for solving differential equations. PDEs have become popular and useful tools for image denoising, image segmentation, and image restoration. Such PDE-based methodology has been successfully applied to many problems in image processing and computer vision. Examples include image denoising, segmentation of textures and remotely sensed data, object detection, optical flow, stereo, and enhancing textures such as fingerprints, image sharpening, and target tracking. The basic idea is to use finite differences to approximate those differentials in the PDEs. Due to its simplicity, the difference method is often the first choice for those who are interested in numerical solutions of PDEs. One disadvantage of this method is that it becomes quite complex when solving PDEs on irregular domains. The finite volume method is another method for solving PDEs.