The β-Delaunay tessellation in R d−1 is a generalization of the classical Poisson-Delaunay tessellation. As a first result of this paper we show that the shape of a weighted typical cell of a β-Delaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex in R d−1 . This generalizes earlier results of Hug and Schneider about the typical (non-weighted) Poisson-Delaunay simplex. Second, the asymptotic behaviour of the volume of weighted typical cells in high-dimensional β-Delaunay tessellation is analysed, as d → ∞. In particular, various high dimensional limit theorems, such as quantitative central limit theorems as well as moderate and large deviation principles, are derived.