  This is an elementary review, aimed at non-specialists, of results that have
been obtained for the limiting distribution of eigenvalues and for the operator
norms of real symmetric random matrices via the method of moments. This method
goes back to a remarkable argument of Eugen Wigner some sixty years ago which
works best for independent matrix entries, as far as symmetry permits, that are
all centered and have the same variance. We then discuss variations of this
classical result for ensembles for which the variance may depend on the
distance of the matrix entry to the diagonal, including in particular the case
of band random matrices, and/or for which the required independence of the
matrix entries is replaced by some weaker condition. This includes results on
ensembles with entries from Curie-Weiss random variables or from sequences of
exchangeable random variables that have been obtained quite recently.
