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