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Polygon meshes, which are used in most graphics applications, require considerable amounts of storage, even when they only approximate precise shapes with limited accuracy. To support Internet access to 3D models of complex virtual environments or assemblies for electronic shopping, collaborative CAD, multi-player video games, and scientific visualization, representations of 3D shapes must be compressed by several orders of magnitude. Furthermore, several closely related methods have been proposed in recent years to smooth, de-noise, edit, compress, transmit, and animate very large polygon meshes, based on topological and combinatorial methods, signal processing techniques, constrained energy minimization, and the solution of diffusion differential equations. This is an overview of some of my recent results in this area: linear anisotropic mesh filtering, bi-level isosurface compression, space-optimized texture maps, and volume warping for adaptive isosurface extraction.