  We discuss and prove validity of the Hubbard-Stratonovich (HS) identities
over hyperbolic domains which are used frequently in the studies on disordered
systems and random matrices. We also introduce a counterpart of the HS identity
arising in disordered systems with "chiral" symmetry. Apart from this we
outline a way of deriving the nonlinear $\sigma$-model from the gauge-invariant
Wegner $k-$orbital model avoiding the use of the HS transformations.
