Digital geometry processing is a technique about how to handle point-cloud data or mesh data, which is an important branch of computer graphics and has significant application in industry such as simulation and movie. Its main research region includes surface reconstruction, simplification, fairing, parameterization, mesh generation, mesh compression, mesh editing and many more. It also has close relation with animation and rendering. However, digital geometry data has several characters which make it difficult to process, such as non-uniform sampling, non-regular mesh, topology complexity, lack of volume information, large noise, etc. Many open problems exist in this active research area. In this thesis, we study several problems about digital geometry processing. The details lie in two aspects: cross-parameterization among surfaces and intrinsic scale-space on manifold.