The most important result of this research has been the development of exploitation of the connection between nonlinear evolution equations solvable by some Inverse Scattering Transform (IST) and ordinary differential equations of Painleve-type. This connection provides a test (or series of tests) which a given evolution equation must pass if it is solvable by IST. Being a necessary condition, it complements the various sufficient conditions that already were known. Independently, the connection also can be used to obtain global information about the classical Painleve transcendents. This information becomes increasingly important as these equations are discovered more and more frequently in models of physical problems. (Author)