  This paper deals with the quantitative normal approximation of non-linear
functionals of Poisson random measures, where the quality is measured by the
Kolmogorov distance. Combining Stein's method with the Malliavin calculus of
variations on the Poisson space, we derive a bound, which is strictly smaller
than what is available in the literature. This is applied to sequences of
multiple integrals and sequences of Poisson functionals having a finite chaotic
expansion. This leads to new Berry-Esseen bounds in de Jong's theorem for
degenerate U-statistics. Moreover, geometric functionals of intersection
processes of Poisson $k$-flats, random graph statistics of the Boolean model
and non-linear functionals of Ornstein-Uhlenbeck-L\'evy processes are
considered.
