We consider the Voronoi tessellation based on a homogeneous Poisson point process in R d . For a geometric characteristic of the cells (e.g. the inradius, the circumradius, the volume), we investigate the point process of the nuclei of the cells with large values. Conditions are obtained for the convergence in distribution of this point process of exceedances to a homogeneous compound Poisson point process. We provide a characterization of the asymptotic cluster size distribution which is based on the Palm version of the point process of exceedances. This characterization allows us to compute efficiently the values of the extremal index and the cluster size probabilities by simulation for various geometric characteristics. The extension to the Poisson-Delaunay tessellation is also discussed.where #S denotes the cardinality of any finite set S. Thanks to the stationarity of m, the intensity does not depend on the choice of A. Without loss of generality, we assume that γ m = 1.The typical cell C of a stationary tessellation m is a random polytope with distribution given by(1.1)where f : K d → R is any bounded measurable function on the set of convex bodies K d (endowed with the Hausdorff topology). Let χ be a locally finite subset of R d . The Voronoi cell with nucleus x ∈ χ is the set of all sites y ∈ R d whose distance from x is smaller or equal than the distances to all other points of χ, i.e. C χ (x) := {y ∈ R d : |y − x| ≤ |y − x |, x ∈ χ}. When χ = η is a homogeneous Poisson point process, the family m := {C η (x) : x ∈ η} is the so-called Poisson-Voronoi tessellation. The intensity of such a tessellation equals the intensity of η. A consequence of the theorem of Slivnyak (see e.g. Theorem 3.3.5 in [28]) shows that C