  Hitchin's twistor treatment of Schlesinger's equations is extended to the
general isomonodromic deformation problem. It is shown that a generic linear
system of ordinary differential equations with gauge group SL(n,C) on a Riemann
surface X can be obtained by embedding X in a twistor space Z on which sl(n,C)
acts. When a certain obstruction vanishes, the isomonodromic deformations are
given by deforming X in Z. This is related to a description of the deformations
in terms of Hamiltonian flows on a symplectic manifold constructed from affine
orbits in the dual Lie algebra of a loop group.
