A number of special cases of systems of ordinary and partial differential equations generating deformations of a rational covariant derivative operator that preserve its generalized monodromy data are known to have a Hamiltonian structure. These include the Schlesinger systems, the Painlev equations, and the case of an operator of arbitrary rank with an arbitrary number of regular singular points, plus a single one with Poincare index 1. The underlying Hamiltonian structure in all these cases has been shown to coincide with the rational R-matrix structure used in the theory of integrable systems. The Hamiltonians are known to coincide with the logarithmic derivatives of the associated isomonodromic tau functions, and also to be spectral invariants of the associated rational connection matrix. These results will be extended to include the general, non-resonant rational isomonodromic deformation system on a punctured Riemann sphere, as well as systems over elliptic curves. If time permits, it will be shown how specific classes of solutions satisfying ”Virasoro constraints” may be produced from the theory of generalized semi-classical orthogonal polynomials. The tau function for these solutions is explicitly related to the partition function of an associated random matrix model.