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In this paper two new classes of stationary random simplicial tessellations,the so-called $\beta$- and $\beta'$-Delaunay tessellations, are introduced.Their construction is based on a space-time paraboloid hull process andgeneralizes that of the classical Poisson-Delaunay tessellation. Thedistribution of volume-power weighted typical cells is explicitly identified,establishing thereby a remarkable connection to the classes of $\beta$- and$\beta'$-polytopes. These representations are used to determine principalcharacteristics of such cells, including volume moments, expected angle sumsand cell intensities.

The $\beta$-Delaunay tessellation I: Description of the model and geometry of typical cells