  These lectures present a survey of recent developments in the area of random
matrices (finite and infinite) and random permutations. These probabilistic
problems suggest matrix integrals (or Fredholm determinants), which arise very
naturally as integrals over the tangent space to symmetric spaces, as integrals
over groups and finally as integrals over symmetric spaces. An important part
of these lectures is devoted to showing that these matrix integrals, upon
apropriately adding time-parameters, are natural tau-functions for integrable
lattices, like the Toda, Pfaff and Toeplitz lattices, but also for integrable
PDE's, like the KdV equation. These matrix integrals or Fredholm determinants
also satisfy Virasoro constraints, which combined with the integrable equations
lead to (partial) differential equations for the original probabilities.
