  In this paper we apply to the zeros of families of $L$-functions with
orthogonal or symplectic symmetry the method that Conrey and Snaith used to
calculate the $n$-correlation of the zeros of the Riemann zeta function. This
method uses the Ratios Conjectures for averages of ratios of zeta or
$L$-functions. Katz and Sarnak conjecture that the zero statistics of families
of $L$-functions have an underlying symmetry relating to one of the classical
compact groups $U(N)$, $O(N)$ and $USp(2N)$. Here we complete the work already
done with $U(N)$ to show how new methods for calculating the $n$-level
densities of eigenangles of random orthogonal or symplectic matrices can be
used to create explicit conjectures for the $n$-level densities of zeros of
$L$-functions with orthogonal or symplectic symmetry, including all the lower
order terms. We show how the method used here results in formulae that are
easily modified when the test function used has a restricted range of support,
and this will facilitate comparison with rigorous number theoretic $n$-level
density results.
