Recently, Katz and Sarnak (inspired in part by former work of Montgomery) have proposed a probabilistic model for the distribution of the zeros of $L$ functions in terms of eigenvalues of random matrices of large rank. In this lecture, we list the various statistics that have been used to test this model (without any contradiction in sight so far !), and in particular the level one distribution of “low lying zeros” in families of automorphic $L$ functions studied in details by Iwaniec, Luo and Sarnak. We also discuss several important applications which were inspired by this model.