  We introduce random matrix ensembles that correspond to the infinite families
of irreducible Riemannian symmetric spaces of type I. In particular, we recover
the Circular Orthogonal and Symplectic Ensembles of Dyson, and find other
families of (unitary, orthogonal and symplectic) ensembles of Jacobi type. We
discuss the universal and weakly universal features of the global and local
correlations of the levels in the bulk and at the hard edge of the spectrum
(i.e., at the "central points" +/-1 on the unit circle). Previously known
results are extended, and we find new simple formulas for the Bessel Kernels
that describe the local correlations around a central point.
