  Strongly non-Gaussian ensembles of large random matrices possessing unitary
symmetry and logarithmic level repulsion are studied both in presence and
absence of hard edge in their energy spectra. Employing a theory of polynomials
orthogonal with respect to exponential weights we calculate with asymptotic
accuracy the two-point kernel over all distance scale, and show that in the
limit of large dimensions of random matrices the properly rescaled local
eigenvalue correlations are independent of level confinement while global
smoothed connected correlations depend on confinement potential only through
the endpoints of spectrum. We also obtain exact expressions for density of
levels, one- and two-point Green's functions, and prove that new universal
local relationship exists for suitably normalized and rescaled connected
two-point Green's function. Connection between structure of Szeg\"o function
entering strong polynomial asymptotics and mean-field equation is traced.
