  This paper deals with U-statistics of Poisson processes and multiple
Wiener-It\^o integrals on the Poisson space. Via sharp bounds on the cumulants
for both classes of random variables, moderate deviation principles,
concentration inequalities and normal approximation bounds with Cram\'er
correction are derived. It is argued that the results obtained in this way are
in a sense best possible and cannot be improved systematically. Applications in
stochastic geometry and to functionals of Ornstein-Uhlenbeck-L\'evy processes
are investigated.
