Jonathan Shewchuk (University of California, Berkeley, U.S.A.) Many tasks in scientific computing and computer graphics require surface meshes or volume meshes composed of high-quality triangles or tetrahedra. Some mesh generation algorithms use the mathematical properties of Delaunay triangulations to offer guarantees on the quality of the meshes they produce. I discuss mesh generation algorithms that take advantage of two variants of the Delaunay triangulation to resolve specific problems. Three-dimensional domains whose polygonal boundaries meet at small angles are particularly difficult to mesh with high-quality tetrahedra; weighted Delaunay triangulations provide a way to ensure that a mesh will conform to the shape of the domain. Curved surfaces embedded in three dimensions are difficult to mesh because of the difficulty of ensuring that a piecewise linear mesh will be a topologically and geometrically accurate representation of a surface; restricted Delaunay triangulations, coupled with a theory of surface sampling, provide a way to guarantee this accuracy along with a guarantee of high-quality triangles. The theory and algorithms in this talk will appear in a forthcoming book, "Delaunay Mesh Generation", by Siu-Wing Cheng, Tamal Dey, and myself.