  In this article we obtain concentration inequalities for Poisson
$U$-statistics $F_m(f,\eta)$ of order $m\ge 1$ with kernels $f$ under general
assumptions on $f$ and the intensity measure $\gamma \Lambda$ of underlying
Poisson point process $\eta$. The main result are new concentration bounds of
the form \[
  \mathbb{P}(|F_m ( f , \eta) -\mathbb{E} F_m ( f , \eta)| \ge t)\leq
2\exp(-I(\gamma,t)), \]
  where $I(\gamma,t)$ satisfies $I(\gamma,t)=\Theta(t^{1\over m}\log t)$ as
$t\to\infty$ and $\gamma$ is fixed. The function $I(\gamma,t)$ is given
explicitly in terms of parameters of the assumptions satisfied by $f$ and
$\Lambda$. One of the key ingredients of the proof are fine bounds for the
centred moments of $F_m(f,\eta)$. We discuss the optimality of obtained bounds
and consider a number of applications related to Gilbert graphs and Poisson
hyperplane processes in constant curvature spaces.
