I will start by recalling some known concentration estimates and limit theorems for Poisson point processes with focus on U-statistics. Next I will discuss moment and tail inequalities for multiple stochastic integrals (of deterministic functions), compare them with analogous inequalities on the Gauss space (due to Latała) and classical U-statistics and show how they can be used to derive the law of the iterated logarithm when the intensity of the process tends to infinity. I will also discuss similarities and differences with the law of the iterated logarithm for classical U-statistics. Finally, I will illustrate these results with specific examples from stochastic geometry and stochastic processes. I will conclude with some open problems. Based on joint work with Dominik Kutek (University of Warsaw).