Abstract: Delaunay triangulations are subdivisions of a domain into triangles obeying certain geometric constraints. These have use in many applications such as mesh generation, finite element analysis, and interpolation. In this talk, I will discuss the Delaunay triangulation and its computation, including topics such as anisotropic mesh refinement and curve-bounded domains that are important for obtaining high-quality discretisations. I will briefly cover the related Voronoi tessellation. Throughout this discussion I will demonstrate how my Julia package, DelaunayTriangulation.jl, can be used to perform these computations efficiently and applied to solving partial differential equations and to interpolation.