Although the first appearence of symplectic manifolds was in physics, they have played a fundamental role in pure math for the last 30 years. A big breakthrough came after the work of Mikhail Gromov, who started the study of pseudoholomorphic curves in symplectic manifolds. This make a connection with ideas coming from algebraic geometry. The same kind of ideas allowed Andreas Floer to introduce Floer's homology which has revolutionized low dimensional topology in the last 20 years. I plan to talk about topological properties of those manifolds and the basic facts about the moduli space of pseudoholomorphic curves in a symplectic manifold and how, once we have constructed it, this gives amazing applications to geometry. I'm also giving a proof of the nonsqueezing theorem of Gromov.