In this talk we give a short introduction to the Ziglin-Morales-Ramis approach to the problem of meromorphic integrability of complex Hamiltonian Systems. This approach lies at the crossroads of differential Galois theory, symplectic geometry and variational methods. In the last 20 years this theory has been crucial to effectively prove through computer algebra methods the non integrability of a number of dynamical systems that had resisted other powerful and sophisticated approaches. Another beautiful aspect of this theory, which trascends its concrete usefulness, is its mixed nature showing that all fields of mathematics are ultimately interlinked, and how numerical computation and computer algebra can be complementary.
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