Inpainting, or image interpolation, has broad applications in visual perception, digital media, and information technology. Compared with other classical interpolation problems (such as polynomial-, spline-, Shannon- or wavelet-interpolations), inpainting imposes extra challenges mainly due to the complexity of both missing domains and missing signals, especially the geometric features of images. In this talk, we focus on our recent efforts in applying the variational-PDE approach to inpainting, and reveal novel intriguing applications of nonlinear PDEs, geometric measure theory, and Gamma-convergence in mathematical imaging and vision. (Joint work with several authors in a series of papers, especially with the applied math group at UCLA.)