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The β-Delaunay tessellation in ℝd-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 βDelaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex inℝd-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. [ABSTRACT FROM AUTHOR]