  We summarize the recent results about complete solvability of Hermitian and
rectangular complex matrix models. Partition functions have very simple
character expansions with coefficients made from dimensions of representation
of the linear group $GL(N)$, and arbitrary correlators in the Gaussian phase
are given by finite sums over Young diagrams of a given size, which involve
also the well known characters of symmetric group. The previously known
integrability and Virasoro constraints are simple corollaries, but no vice
versa: complete solvability is a peculiar property of the matrix model
(hypergeometric) $\tau$-functions, which is actually a combination of these two
complementary requirements.
