This article presents a new class of wavelets, based on subdivision surfaces, that extends the class of representable functions to include functions defined on compact surfaces of arbitrary topological type. The authors introduce "subdivision wavelets," which can be used for various applications such as continuous level-of-detail control for graphics rendering, compression of geometric models, and acceleration of global illumination algorithms. The article provides a theoretical foundation for multiresolution analysis on surfaces of arbitrary topological type, including detailed discussions on nested linear spaces through subdivision, inner products, and the construction of wavelets. The methods described are applicable to a wide range of subdivision schemes, including primal and dual schemes, and are capable of representing smooth multiresolution surfaces of arbitrary topological type. The authors also discuss the practical implications of their work, such as efficient wavelet compression methods and the ability to achieve smooth variation in detail levels.This article presents a new class of wavelets, based on subdivision surfaces, that extends the class of representable functions to include functions defined on compact surfaces of arbitrary topological type. The authors introduce "subdivision wavelets," which can be used for various applications such as continuous level-of-detail control for graphics rendering, compression of geometric models, and acceleration of global illumination algorithms. The article provides a theoretical foundation for multiresolution analysis on surfaces of arbitrary topological type, including detailed discussions on nested linear spaces through subdivision, inner products, and the construction of wavelets. The methods described are applicable to a wide range of subdivision schemes, including primal and dual schemes, and are capable of representing smooth multiresolution surfaces of arbitrary topological type. The authors also discuss the practical implications of their work, such as efficient wavelet compression methods and the ability to achieve smooth variation in detail levels.