  Universality of local eigenvalue statistics is one of the most striking
phenomena of Random Matrix Theory, that also accounts for a lot of the
attention that the field has attracted over the past 15 years. In this paper we
focus on the empirical spacing distribution and its Kolmogorov distance from
the universal limit. We describe new results, some analytical, some numerical,
that are contained in [27]. A large part of the paper is devoted to explain
basic definitions and facts of Random Matrix Theory, culminating in a sketch of
the proof of a weak version of convergence for the empirical spacing
distribution $\sigma_N$.
