  The $\beta$-Delaunay tessellation in $\mathbb{R}^{d-1}$ 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 $\beta$-Delaunay
tessellation, conditioned on having large volume, is close to the shape of a
regular simplex in $\mathbb{R}^{d-1}$. 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 $\beta$-Delaunay tessellation is analysed, as $d\to\infty$. In
particular, various high dimensional limit theorems, such as quantitative
central limit theorems as well as moderate and large deviation principles, are
derived.
