We prove tail and moment inequalities for multiple stochastic integrals on the Poisson space and for Poisson $U$-statistics. We use them to demonstrate the Law of the Iterated Logarithm for these processes when the intensity of the Poisson process tends to infinity, with normalization depending on the degree of the multiple stochastic integral or degeneracy of the kernel defining the $U$-statistic. We apply our results to several classical functionals of Poisson point processes, obtaining improvements or complements of known concentration of measure results as well as new laws of the iterated logarithm. Examples include subgraph counts and power length functionals of geometric random graphs, intersections of Poisson $k$-flats, quadratic functionals of the Ornstein--Uhlenbeck Lévy process and $U$-statistics of marked processes. Keywords: Poisson point process, $U$-statistics, multiple stochastic integrals, concentration of measure, The Law of the Iterated Logarithm