Abstract Over the last three decades, wavelet theory has been developed as a powerful mathematical tool to extract information from many kinds of data such as audio signal and images. Compared to global bases such as the Fourier base, wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. Another main advantage is that wavelets can form a multiresolution analysis (MRA) of interested function spaces. This means that we can perform the multiresolution analysis for data. Wavelets have been widely used to process scalar fields in the fields of Computer Graphics, Computer Vision and Image Processing, such as image processing (denoising, compression and fusion), geometry processing (denoising, compression), and rendering (adaptive rendering, compression). However, there are very few works that apply wavelets to process vector fields of 2D and 3D. In Computer Graphics, there are many important problems associated with vector fields such as gradient-domain image processing, surface reconstruction from oriented points, and fluid simulation. If we still use traditional wavelet methods to process vector fields by representing them as scalar fields, we would miss some nice properties of vector fields. According to the famous Helmholtz-Hodge decomposition theorem, any vector field can be decomposed into the sum of a curl-free field, a divergence-free field and a harmonic field (both curl-free and divergence-free). We know that the set of each kind of the vector fields forms a subspace. It would be better to construct wavelets for those subspaces rather than using wavelets for scalar fields. In this thesis, we develop two kinds of powerful vector-valued wavelets for processing vector fields called curl-free and divergence-free wavelets. The constructed curl-free wavelets (resp. divergence-free wavelets) form a basis of the curl-free subspace (resp. divergence-free subspace). With the two kinds of wavelets, we design efficient algorithms for solving three important problems in Computer Graphics: Gradient-domain image compositing, Surface reconstruction from oriented points, and Fluid simulation. For the problem of Gradient-domain image compositing, we develop an efficient approximate wavelet algorithm for gradient-domain image compositing on both the CPU an GPU. The algorithm has O(n) time complexity and 2n memory cost for an n-pixels composition and outperforms state-of-the-art methods both on the CPU and GPU. For the problem of Surface reconstruction, we develop a new biorthogonal wavelet approach to creating a water-tight surface from oriented point sets. The approach supports streaming implementations for processing very large datasets. For the problem of Fluid simulation, we develop a new multiresolution wavelet algorithm for solving the Helmholtz-Hodge decomposition problem and apply it to smoke simulation and turbulence synthesis. Keywords: Computer Graphics, Curl-free Wavelets, Divergence-free Wavelets, Surface Reconstruction, Gradient-domain image processing, Fluid Simulation, Turbulence Processing