In this talk, we present some results originating from a new general inequality obtained by combining the Chen–Stein method with Malliavin calculus on the Poisson space, such as multidimensional Poisson approximations, mixed limit theorems, as well as a characterization of asymptotic independence for U–statistics. Applications to stochastic geometry through limit theorems involving the joint convergence of vectors of subgraph–counting statistics exhibiting both a Poisson and a Gaussian behavior will also be discussed.