  We derive exact results for gap probabilities, as well as densities of
extreme eigenvalues for six complex random matrix ensembles of fundamental
importance. These are Gauss-Wigner, Laguerre-Wishart, Cauchy-Lorentz (two
variants), Jacobi-MANOVA and Bures-Hall (trace unrestricted). We deal with both
correlated and uncorrelated cases. Extensive Monte-Carlo simulations based on
explicit matrix models or Dyson's log-gas formalism have also been performed,
which corroborate all analytical results.
