  We give a proof of the Universality Conjecture for orthogonal (beta=1) and
symplectic (beta=4) random matrix ensembles of Laguerre-type in the bulk of the
spectrum as well as at the hard and soft spectral edges. Our results are stated
precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5,
1.7). They concern the appropriately rescaled kernels K_{n,beta}, correlation
and cluster functions, gap probabilities and the distributions of the largest
and smallest eigenvalues. Corresponding results for unitary (beta=2)
Laguerre-type ensembles have been proved by the fourth author in [23]. The
varying weight case at the hard spectral edge was analyzed in [13] for beta=2:
In this paper we do not consider varying weights.
  Our proof follows closely the work of the first two authors who showed in
[7], [8] analogous results for Hermite-type ensembles. As in [7], [8] we use
the version of the orthogonal polynomial method presented in [25], [22] to
analyze the local eigenvalue statistics. The necessary asymptotic information
on the Laguerre-type orthogonal polynomials is taken from [23].
