  While Random Matrix Theory has successfully modeled many quantities of
families of L-functions, it frequently cannot see the family's arithmetic. In
some situations this requires an extended theory that inserts arithmetic
factors depending on the family, while in other cases these factors result in
contributions which vanish in the limit, and are thus not detected. We review
the general theory associated to one of the most important statistics, the
n-level density of zeros near the central point. According to the Katz-Sarnak
density conjecture, to each family of L-functions there is a corresponding
symmetry group such that the behavior of zeros near the central point as the
conductors tend to infinity agrees with the behavior of eigenvalues near 1 as
the matrix size tends to infinity. We show how these calculations are done,
emphasizing the techniques, methods and obstructions to improving the results,
by considering in full detail a family of Dirichlet characters. We then
describe how we may associate a symmetry constant to each family, and how to
determine the symmetry group of a compound family in terms of the symmetries of
the constituents. These calculations explain the remarkable universality of
behavior, where the main terms are independent of the arithmetic (only the
first two moments of the Satake parameters contribute in the limit; similar to
the Central Limit Theorem, the higher moments are only felt in the rate of
convergence). We end by exploring lower order terms in families of elliptic
curves. We present evidence supporting a conjecture that the average second
moment in one-parameter families without complex multiplication has, when
appropriately viewed, a negative bias, and end with a discussion of the
consequences of this bias on the distribution of low-lying zeros, in particular
relations between such a bias and the observed excess rank in families.
