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These notes are based on a course that took place at the Universit\'e deNantes in June 2011 during the "Trimester on Contact and Symplectic Topology".We will explain how holomorphic curves can be used to study symplectic fillingsof a given contact manifold. Our main goal consists in showing that certaincontact manifolds do not admit any symplectic filling at all.

Higher dimensional contact topology via holomorphic disks