  Let $X$ be the mosaic generated by a stationary Poisson hyperplane process
$\hat X$ in ${\mathbb R}^d$. Under some mild conditions on the spherical
directional distribution of $\hat X$ (which are satisfied, for example, if the
process is isotropic), we show that with probability one the set of cells
($d$-polytopes) of $X$ has the following properties. The translates of the
cells are dense in the space of convex bodies. Every combinatorial type of
simple $d$-polytopes is realized infinitely often by the cells of $X$. A
further result concerns the distribution of the typical cell.
