  We provide a comprehensive analysis of matrix-valued Herglotz functions and
illustrate their applications in the spectral theory of self-adjoint
Hamiltonian systems including matrix-valued Schr\"odinger and Dirac-type
operators. Special emphasis is devoted to appropriate matrix-valued extensions
of the well-known Aronszajn-Donoghue theory concerning support properties of
measures in their Nevanlinna-Riesz-Herglotz representation. In particular, we
study a class of linear fractional transformations M_A(z) of a given n \times n
Herglotz matrix M(z) and prove that the minimal support of the absolutely
continuos part of the measure associated to M_A(z) is invariant under these
linear fractional transformations.
  Additional applications discussed in detail include self-adjoint finite-rank
perturbations of self-adjoint operators, self-adjoint extensions of densely
defined symmetric linear operators (especially, Friedrichs and Krein
extensions), model operators for these two cases, and associated realization
theorems for certain classes of Herglotz matrices.
