Meromorphic nonintegrability of Hamiltonian systems

We give a review of some nonintegrability criteria for complex analytic Hamiltonian systems. These results were obtained by J. P. Ramis and the author, and they are generalizations of Ziglinʹs nonintegrability theorem about the monodromy group of the first order variational equation around a particular solution. We also state a recent generalization of the above to the higher order variational equations.