  Integrable theory is formulated for correlation functions of characteristic
polynomials associated with invariant non-Gaussian ensembles of Hermitean
random matrices. By embedding the correlation functions of interest into a more
general theory of tau-functions, we (i) identify a zoo of hierarchical
relations satisfied by tau-functions in an abstract infinite-dimensional space,
and (ii) present a technology to translate these relations into hierarchically
structured nonlinear differential equations describing the correlation
functions of characteristic polynomials in the physical, spectral space.
Implications of this formalism for fermionic, bosonic, and supersymmetric
variations of zero-dimensional replica field theories are discussed at length.
A particular emphasis is placed on the phenomenon of fermionic-bosonic
factorisation of random-matrix-theory correlation functions.
