In this talk, I will give an overview over some recent asymptotic results for stabilizing functionals of point processes. I will start to explain how Poisson and normal approximation in a strong metric (total variation, Kantorovich-Rubinstein distance, Kolmogorov distance) for functionals of a Poisson process can be obtained by an explicit coupling with the Palm measure. Thereafter, we will consider approximation results in weaker metrics for a larger class of point processes satisfying a spatial mixing assumption. Examples from random graph theory will illustrate the results.